Labour is ready to whip its MPs to back a second Brexit referendum in crucial votes over the coming days, in what backers said was a “significant” step forward in the push for a public vote.

The confirmation from the shadow Brexit secretary, Sir Keir Starmer, that Labour will allow Boris Johnson’s Brexit deal through parliament if it is subject to a confirmatory referendum came after up to 1 million people took to the streets of London to demand a Final Say.

And in just 24 hours after the giant People’s Vote march, more than 200,000 signed a letter to MPs, MEPs and European leaders urging them to ensure that any Brexit deal is put back to the British people before coming into effect.

The Letter from the People to the Powerful, backed by The Independent, states that the deal reached last week between Mr Johnson and EU27 leaders “cannot meet the promises made [in the 2016 referendum campaign], nor be said to represent the will of the people”.

Mr Johnson will on Monday make a second attempt to win MPs’ approval for his deal in a “meaningful vote”, after his first bid on Saturday was thwarted by a 322-306 vote in favour of withholding the green light until all ratification legislation was complete.

But the speaker, John Bercow, is expected to reject the motion, after saying that it would be “quite curious and irregular” to allow a vote intended purely to invalidate Saturday’s result.

And sources close to the group of former Tories expelled from the party for rebelling against no deal said that if a meaningful vote was called, it would be countered with a similar amendment to the one tabled by Sir Oliver Letwin that forced Mr Johnson to write a letter to Brussels asking for an extension to the Brexit process.

Dominic Raab, the foreign secretary, said he believed the government could get its deal through parliament. “We seem to have the numbers in the House of Commons. A lot of people say ‘Get this done and move on’,” he said.

Mr Johnson’s unsigned letter – accompanied by a signed message advising against a “corrosive” delay – was described as a potential contempt of parliament by Labour’s John McDonnell and will be the subject of legal action in Scotland’s highest court on Monday.

SNP MP Joanna Cherry, who led the successful legal challenge to the PM’s unlawful suspension of parliament, branded his letters “pathetic” and confirmed she will pursue action in Edinburgh’s Court of Session.

There was no sign of an immediate extension offer from the EU, with chief negotiator Michel Barnier due to meet the European parliament on Monday to discuss its ratification process.

MPs voted on Saturday to withhold approval of Boris Johnson’s deal (PRU/AFP/Getty)

The government is also due to table its Withdrawal Agreement Bill, with the crucial second reading vote on Tuesday giving MPs the chance to throw out Mr Johnson’s deal or allow it to go forward for debate and possible amendment.

Speaking on BBC1’s Andrew Marr Show, Sir Keir said any referendum amendment was likely to be tabled by backbench MPs rather than the Labour leadership, in order to encourage the broadest possible cross-party support. The party is expected to focus on amendments to boost ties with the EU’s customs union and to prevent a no-deal cliff-edge at the end of the proposed transition period in December 2020.

The Final Say movement is increasingly hopeful of winning a Commons vote, with one source saying that a division on a referendum between Remain and Mr Johnson’s plan was now “on a knife-edge”, and one pitching Remain against no deal was comfortably winnable, in the event that the PM’s deal fails.

“Of course we need an amendment to say that whatever deal gets through, it should be subject to a referendum where that deal is put to the public and they’re asked, ‘do you want to leave on these terms, or would you rather remain in the EU’,” said Sir Keir.

Asked about the crucial Labour whip, he added: “We’ve already voted, I think, three times as a party for a second referendum with a three-line whip behind it. That’s the clear policy.”

Shadow cabinet members Sir Keir Starmer, Diane Abbott, John McDonnell and Emily Thornberry called for a Final Say vote at Saturday’s march (AFP)

It “makes sense” that Labour would repeat the offer made to Theresa May when it said “we don’t think your deal is very good, but if it’s up against the safeguard of being able to remain, then we will allow it to proceed in that way”, he said.

Former foreign secretary Margaret Beckett, a backer of a people’s vote, said Sir Keir’s words were “significant”.

“It is clear now that the party as a whole and the shadow cabinet are firmly behind a referendum in a way that people weren’t sure of before,” she told The Independent. “There’s no doubt that quite a number of people who were very much opposed have come round to the idea.”

Labour’s Peter Kyle said he would table an amendment for a confirmatory referendum “as soon as politically appropriate”, though timing was uncertain because of the unpredictability of the government’s plans.

An “indicative vote” on a second referendum moved by Mr Kyle and fellow Labour MP Phil Wilson failed by just 12 votes earlier this year. But an amendment may struggle to improve on this result, as 45 Tories – including a number of government ministers – abstained in that division but could be expected to oppose it this time around.

Key to success would be winning over pro-deal Labour MPs and exiled Tories sitting as independents. Most of these are likely to be swayed only if Tuesday’s vote is close enough to suggest that Mr Johnson’s package needs a referendum attached to be sure of passing.

Up to a million people took to the streets of London on Saturday to demand a Final Say (Angela Christofilou/The Independent)

And there are also tentative feelers being put out to the Democratic Unionist Party after its Westminster leader Nigel Dodds suggested it was looking at options to amend the PM’s deal, which they regard as unacceptable.

The party’s Brexit spokesperson, Sammy Wilson, said the DUP was not seeking a second referendum.

But Sir Keir said: “I would openly invite the DUP to talk to us, because anybody who wants to improve on the situation – and they do – should be working together.”

A pro-referendum source said the DUP might be swayed by the thought that they are faced with a choice between a general election which they do not relish – if Mr Johnson seeks to capitalise on his deal going through – or a referendum.

Sir Keir acknowledged that staging a fresh public vote would take as long as 22 weeks. But he said the PM was trying to hurry the process through in order to avoid having the flaws in his plan uncovered.

“The more scrutiny there is of Boris Johnson’s deal, the more I think people are going to be persuaded that it’s a very bad deal,” he said. “And one of the reasons I think he’s trying to bounce us through this is because if you look at the detail it exposes his half-truths and untruths.”

But a Downing Street source said: “We cannot allow parliament’s letter to lead to parliament’s delay – we must leave on 31 October and finally get Brexit done. The best way of doing this is for MPs to vote for Boris Johnson’s new deal.”

It is expected that the debate on the Withdrawal Agreement Bill will involve long hours in the Commons and potential weekend sittings in the Lords.

A government source said: “The public will expect us to move at pace to get the withdrawal agreement passed so we can leave on 31 October. There will be time for parliament to consider the legislation and any amendments.”

Meanwhile, the former deputy prime minister Michael Heseltine urged voters to sign the Letter from the People to the Powerful to put maximum pressure on politicians to give the public a say.

“One million supporters of the people’s vote marched outside parliament, while inside MPs from all parties stood firm against Boris Johnson’s efforts to force his hard and bitter Brexit on our country,” said Lord Heseltine.

“From the Highlands, to Belfast, from Cardiff to Penzance, people came from across every corner of the land to make themselves heard.

“We will march on. We’re sending a letter from the people to the powerful to demand they don’t turn their backs and instead give the people the final say. Sign the letter. Send the message. Let’s secure a people’s vote.”