A majority of voters in all seats held by Labour support a second referendum on Brexit, according to an analysis released by the People’s Vote campaign as it steps up its lobbying of opposition MPs.

The campaign commissioned a YouGov study based on polling of 26,000 people in the hope of persuading Labour MPs to vote for a second referendum if May’s Brexit deal is rejected and the party cannot force a general election.

The effort is aimed in particular at Labour MPs in constituencies that supported leave, many of whom still say that the primary message they hear from constituents is they want the UK to get out of the European Union as fast as possible.

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It has helped sign up one high-profile backbencher, Jess Phillips, whose Birmingham Yardley constituency voted 60% to leave the EU but, where, according to the YouGov analysis, 63% support a second referendum and 52% would vote remain.

“It’s no surprise to me that my constituents feel that they have been failed in the negotiations and are changing their mind on Brexit,” Phillips said. “If I had to pick who decided my fate between Theresa May or the people in my community I’d pick them every single time.”

The support of Labour’s 257 MPs will be critical if a second referendum is to pass the House of Commons, and even now strategists at the People’s Vote campaign concede “there is not currently a majority in support of a People’s Vote at Westminster”.

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Their belief is that MPs would first have to vote down whatever deal May brought back from Brussels, before the idea of a second referendum to break the deadlock could gain traction. But the hope is that the research will give Labour MPs food for thought.

Jeremy Corbyn’s party agreed a policy at its conference under which it would measure up May’s deal to see if it would protect jobs and workers’ rights. If the deal failed that test, Labour would vote against it when it was sent to the Commons for ratification after the prime minister concluded the divorce talks with Brussels.

Labour would then seek to force a general election by calling a confidence vote, and only if that failed would it consider the option of a second referendum, an idea that was enthusiastically championed at conference by the shadow Brexit secretary, Sir Keir Starmer.

Support for a second referendum is largest in Corbyn’s seat of Islington North at 80%, with the number falling back to 53% in Ashfield, the east Midlands seat held by Gloria De Piero.

YouGov used a multi-level regression and post-stratification (MRP) technique, polling 26,000 people to build up a statistical model of key demographics, including age, gender, education, race and social class. The theory is that it becomes possible to produce a model for every constituency in the country.

Peter Kellner, a former president of YouGov, said that the MRP approach indicated a hung parliament in last year’s election. He added that his former employer’s data showed that “very few voters with strong views on Brexit – whether for or against – have changed their mind since the referendum”.

Quieter members of the electorate – those “who do not necessarily speak to MPs on the doorstep” – were the ones who had changed their minds, Kellner added.