A majority of people would rather the UK have a soft Brexit, which sees the country give concessions on EU immigration in return for access to the single market, a poll has found.

The exclusive survey for The Independent by BMG Research revealed that people preferred the option compared to a hard Brexit that sees the UK leave the single market altogether to gain “full control” of its borders.

The data, coming as Conservatives prepare for a battle at their conference over the terms of Britain’s EU withdrawal, flies in the face of the stronger approach backed by Cabinet Brexiteers David Davis, Liam Fox and Boris Johnson.

BMG’s research also confirms that a large majority of people believe Theresa May’s support for the Remain camp during the EU referendum was either weak or passive, following claims by one of David Cameron’s top ex-aides that she refused to properly support the campaign.

Perhaps more important to the Prime Minister will be the poll figures showing voters overwhelmingly see her as more concerned about immigration, the working class and disadvantaged children, and less concerned with her own career, than Mr Cameron.

Overall the poll showed the Tories under Ms May enjoying a substantial lead going into their Birmingham conference, on 38 per cent compared to Labour’s 29 per cent, with Ukip on 15 per cent, the Liberal Democrats on 8 per cent and the Greens on 4 per cent.

BMG asked more than 1,000 people at the end of September whether they prefer a hard Brexit in which the UK “leaves the single market, leaves the customs union and gains full control over all aspects of immigration” or a soft Brexit which sees Britain retain “a degree of access to the single market, in return for giving some concessions to the EU on immigration to the UK”.

In total, a slim majority – 52 per cent – said they would rather Theresa May take the softer option, compared to 48 per cent who backed the hard option.

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The result is an exact mirror image of the score in the actual EU referendum, in which 52 per cent backed leaving, while 48 per cent supported remaining, indicating a cautious approach from some people who voted for Brexit.

It also comes after International Trade Secretary Liam Fox used a speech this week to underline how the UK could become a fully independent member of the World Trade Organisation after leaving the EU, indicating that he backs a hard Brexit.

The speech was delivered amid a drive by other Tories to push Ms May into a more moderate position on immigration and access to the single market.

Last week, extracts were published from a book written by Sir Craig Oliver, claiming that Ms May had done little to back Mr Cameron’s bid to stay in the EU.

The new BMG research suggests many people in the country hold the same view. Respondents were asked to choose one of four words – strongly, passionately, weakly, or passively – to describe Ms May’s support for Remain. Just 12 per cent said “strongly”, while 21 per cent said “passionately”. But a large proportion – 43 per cent – said the then Home Secretary had only supported Remain “weakly”, and a further 24 per cent said she acted “passively”.

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It is an indication of how angry Ms May’s camp was with Sir Craig’s attack that he will be charged the full commercial price of almost £1,500 to attend Tory conference this year.

Ms May has gone to great lengths since coming to power to differentiate herself from her predecessor, sacking many of his supporters from the front bench and adopting policies like grammar schools that Mr Cameron eschewed.

The claim in her speech launching the education policy, that she will act for the “working class”, appears to have had some impact. Asked to choose between the former and current prime minister on who they thought was more “concerned about working class people”, 70 per cent backed Ms May while only 30 per cent said Mr Cameron.

When asked who was more concerned “about children from poorer backgrounds”, a similar answer emerged with 69 per cent backing Ms May and 31 per cent Mr Cameron.

On immigration, the issue that was arguably the undoing of Mr Cameron, 59 per cent said Ms May was more concerned and 41 per cent said her predecessor.

The question that could really sting Mr Cameron was when people were asked which politician “is/was more concerned with their career”.

Brexit protest: Thousands march in London Show all 12 1 /12 Brexit protest: Thousands march in London Brexit protest: Thousands march in London A woman poses with a home-made European Union flag as Remain supporters gather on Park Lane in London to show their support for the EU in the wake of Brexit PA Brexit protest: Thousands march in London Remain supporters demonstrate in Parliament Square PA Brexit protest: Thousands march in London Tens of thousands of people gathered to protest the result of the EU referendum PA Brexit protest: Thousands march in London A majority of people in the capital voted to remain in the European Union Reuters Brexit protest: Thousands march in London Protesters chanted: “What do we want to do? Stay in the EU” PA Brexit protest: Thousands march in London The march follows a similar rally in Trafalgar Square that was cancelled due to heavy rain – but which tens of thousands of people turned up to anyway Reuters Brexit protest: Thousands march in London Britain voted to leave the European Union in a referendum by 52 per cent to 48 per cent Reuters Brexit protest: Thousands march in London But support for the Leave campaign in urban areas and among young people was significantly lower Rex features Brexit protest: Thousands march in London Marchers gathered at Park Lane at 11am and marched towards Parliament Square PA Brexit protest: Thousands march in London Some protesters held up baguettes in a display of affection for our continental neighbours PA Brexit protest: Thousands march in London The disparity between different parts of the country has promoted a four million signature petition calling for a second referendum and even a renewed push for Scotland to cede from the UK PA Brexit protest: Thousands march in London The event’s organiser, King’s College graduate Kieran MacDermott, wrote: “We can prevent Brexit by refusing to accept the referendum as the final say and take our finger off the self-destruct button" Reuters

In his book titled Unleashing Demons: The Inside Story Of Brexit, Sir Craig, who won his knighthood in Mr Cameron’s controversial resignation Honours List, noted: “Amid the murder and betrayal of the campaign, one figure stayed very still at the centre of it all – Theresa May. Now she is the last one standing”.

But the BMG poll showed 65 per cent of people actually believe it is Sir Craig’s boss who cared more for his own career, compared to just 35 per cent who went for Ms May.