Pro-EU campaigners have drawn up a hit list of 50 Tory MPs who could be persuaded to support a second Brexit referendum after more than 670,000 people took to the streets to demand a vote on Britain’s exit terms.

Five government ministers are among the Conservative politicians deemed ”reachable” by the People’s Vote group, which is ramping up pressure on individual MPs to back a final say on plans to leave the EU.

It comes in the wake of the major demonstration for a fresh Brexit vote on Saturday, organised in partnership with The Independent, which is thought to be the largest march in the capital since the Iraq War protests in 2003.

Only seven Tory MPs have openly backed a second referendum but a significant number are understood to be privately weighing their options as Theresa May struggles to unite both her own party and Brussels on the best way forward.

Former minister Anna Soubry, who was one of the first Conservative MPs to support a new vote, told The Independent: “I can tell you, privately lots and lots of Conservatives MPs are saying, ‘How else are we going to solve this problem? How else are we going to solve the impasse over there in parliament?’

“Remember, [this is] a government that after two and a half years has still not worked out what its policy is and a government that has no mandate for a hard Brexit.”

She was joined by ex-attorney general Dominic Grieve and prominent backbenchers Justine Greening, Sarah Wollaston, Philip Lee, Guto Bebb and Heidi Allen in backing a second referendum.

Dr Lee, who quit the government in June to speak out over Brexit, told activists at Conservative conference that at least three ministers were privately supportive of another referendum.

In a targeted push, the campaigners have set aside more than £100,000 for individual polls of more than 600 constituencies, sometimes at the request of MPs themselves, which are due to be published this week.

Protesters brought the streets to a virtual standstill on Saturday as they marched from Park Lane to Parliament Square, where celebrities including Delia Smith and Deborah Meaden gave speeches, and politicians including Labour’s Chuka Umunna, Tory MP Sarah Wollaston and Liberal Democrat leader Sir Vince Cable addressed the crowds.

Alastair Campbell, Tony Blair’s former spin doctor who attended the march, said voters from both sides of the Brexit debate had been galvanised because they “feel the Brexit process is not going well”.

He told Sky News’ Ridge on Sunday: “I don’t believe there is an ‘it’ [a Brexit deal] that she is going to be able get through parliament. What happens then?

“It may actually be that that is the moment, amid the chaos that might follow that if she can’t get a majority in parliament for a deal, that that’s when it goes to a people’s vote.”

A historic moment: thousands march to demand Final Say on Brexit Show all 65 1 /65 A historic moment: thousands march to demand Final Say on Brexit A historic moment: thousands march to demand Final Say on Brexit An estimated 700,000 people marched through London to demand a final say on the withdrawal agreement Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Red smoke from a canister hangs in the air as around 100,000 demonstrators march through London during a People's Vote anti-brexit demonstration savings banners and placards Anti-Brexit People's Vote March for the Future in London Rex A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Mayor of London Sadiq Khan takes part in the People's Vote March for the Future in London, a march and rally in support of a second EU referendum. PA A historic moment: thousands march to demand Final Say on Brexit Final Say campaigners take part in the peoples vote march for the future in London 20/10/2018 Protesters wearing final Say shirts and holding placards Angela Christofilou A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final say Brexit march - final say on our furture London, 20 October 2018 Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit MP Chuka Umunna (left) and MP Vince Cable (right) as MP Anna Soubry (centre) addresses Anti-Brexit campaigners at a rally after the People's Vote March for the Future in London, a march and rally in support of a second EU referendum. PA A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit A People’s Vote march attendee calls for a Final Say Angela Christofilou A historic moment: thousands march to demand Final Say on Brexit Demonstrators with banners 'We're with EU' during the People's Vote March for the Future in London, Britain, 20 October 2018. Reports state that the 'March for the Future' is to be led by a column of young people and call for a Peopleâ€™s Vote on the Brexit deal. After marching through central London, there will be a rally on stage in Parliament Square, including speeches from Mayor of London Sadiq Khan. EPA A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Protesters at London’s march for the future in October The Independent A historic moment: thousands march to demand Final Say on Brexit Editor of The Independent Christian Broughton speaks to demonstrators in Parliament Sqaure after they take part in a march calling for a People's Vote on the final Brexit deal, in central London on October 20, 2018. - Britons dreading life outside Europe gathered from all corners of the UK to London on Saturday to try to stop their country's looming breakup with the EU. AFP/Getty A historic moment: thousands march to demand Final Say on Brexit A poster at the March for the Future in October The Independent A historic moment: thousands march to demand Final Say on Brexit Anti-Brexit campaigners take part in the People's Vote March for the Future in London, a march and rally in support of a second EU referendum. PA A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say campaigners take part in the peoples vote march for the future in London 20/10/2018 Angela Christofilou A historic moment: thousands march to demand Final Say on Brexit Final say Brexit march - final say on our furture London, 20 October 2018 Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final say Brexit march - final say on our furture London, 20 October 2018 Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Anti-Brexit campaigners take part in the People's Vote March for the Future in London, a march and rally in support of a second EU referendum. PA A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final say Brexit march - final say on our furture London, 20 October 2018 Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Demonstrators hold placards as they take part in a march calling for a People's Vote on the final Brexit deal, in central London on October 20, 2018. - Britons dreading life outside Europe gathered from all corners of the UK to London on Saturday to try to stop their country's looming breakup with the EU. AFP/Getty A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final say Brexit march - final say on our furture London, 20 October 2018 Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final say Brexit march - final say on our furture London, 20 October 2018 Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou A historic moment: thousands march to demand Final Say on Brexit Final say Brexit march - final say on our furture London, 20 October 2018 Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final say Brexit march - final say on our furture London, 20 October 2018 Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final say Brexit march - final say on our furture London, 20 October 2018 Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final say Brexit march - final say on our furture London, 20 October 2018 Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou A historic moment: thousands march to demand Final Say on Brexit Final Say campaigners take part in the peoples vote march for the future in London 20/10/2018 campaigner wrapped in EU flag Angela Christofilou A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say march for the future London, 20 October 2018 Credit: Angela Christofilou/The Independent Angela Christofilou/The Independent Angela Christofilou/The Independent A historic moment: thousands march to demand Final Say on Brexit Final Say campaigners take part in the peoples vote march for the future in London 20/10/2018 Angela Christofilou A historic moment: thousands march to demand Final Say on Brexit Demonstrators pass Trafalgar Square as they take part in a march calling for a People's Vote on the final Brexit deal, in central London on October 20, 2018. - Britons dreading life outside Europe gathered from all corners of the UK to London on Saturday to try to stop their country's looming breakup with the EU. AFP/Getty A historic moment: thousands march to demand Final Say on Brexit Demonstrators wave Union and European flags and hold up placards as they pass Trafalgar Square, taking part in a march calling for a People's Vote on the final Brexit deal, in central London on October 20, 2018. - Britons dreading life outside Europe gathered from all corners of the UK to London on Saturday to try to stop their country's looming breakup with the EU. AFP/Getty A historic moment: thousands march to demand Final Say on Brexit Demonstrators hold placards as they take part in a march calling for a People's Vote on the final Brexit deal, in central London on October 20, 2018. - Britons dreading life outside Europe gathered from all corners of the UK to London on Saturday to try to stop their country's looming breakup with the EU. AFP/Getty A historic moment: thousands march to demand Final Say on Brexit A demonstrator holds a message during a march calling for a People's Vote on the final Brexit deal, in central London on October 20, 2018. - Britons dreading life outside Europe gathered from all corners of the UK to London on Saturday to try to stop their country's looming breakup with the EU. AFP/Getty A historic moment: thousands march to demand Final Say on Brexit Anti-Brexit campaigners take part in the People's Vote March for the Future in London, a march and rally in support of a second EU referendum. PA A historic moment: thousands march to demand Final Say on Brexit Demonstrators take part in the 'People's Vote March for the Future,' in central London, Britain, 20 October 2018. Reports state that the 'March for the Future' is to be led by a column of young people and call for a Peopleâ€™s Vote on the Brexit deal. After marching through central London, there will be a rally on stage in Parliament Square, including speeches from Mayor of London Sadiq Khan EPA A historic moment: thousands march to demand Final Say on Brexit A protester brandishes an Independent t-shirt during the Brexit March Angela Christofilou A historic moment: thousands march to demand Final Say on Brexit Anti-Brexit campaigners take part in the People's Vote March for the Future in London, a march and rally in support of a second EU referendum PA A historic moment: thousands march to demand Final Say on Brexit Tens of thousands of people take part in People's Vote March for the Future in central London. The march organised by the People's Vote campaign is led by young people calling for a People's Vote on the Brexit deal Rex A historic moment: thousands march to demand Final Say on Brexit Anti-Brexit campaigners take part in the People's Vote March for the Future in London, a march and rally in support of a second EU referendum PA

It comes as the prime minister faces a battle to save her leadership in the coming days after a crunch EU summit broke up without any sign of a Brexit deal.

Angry Tory MPs are urging Ms May to make a rare appearance before the 1922 Committee of Conservative backbenchers on Wednesday, to undergo a grilling about her Brexit strategy.

A Downing Street source said a decision had not been made about the prime minister’s diary. It is unusual for any prime minister to address the committee apart from at the beginning and the end of the parliamentary term.