To use a term favoured by sports commentators, we are now reaching the business end of the UK’s Brexit negotiations with the rest of the EU. As we do, public opinion is on the move. This is clear from two huge-scale surveys that have been published in the past few days. Both were conducted by YouGov; both questioned exceptionally large samples; together they help us to understand what is happening. One important finding is that if the Brexit talks break down, a two-to-one majority of voters think the people, not Parliament, should decide whether or not Brexit goes ahead.

The first survey, commissioned jointly by Best for Britain and Hope Not Hate among 15,000 electors, indicates that more than 100 parliamentary constituencies that voted Leave two years ago would now vote Remain. These include exceptionally large swings from Leave to Remain in Labour strongholds, such as Liverpool Walton, Knowsley, Swansea East, Oldham West & Royton – and Hayes & Harlington, whose MP is John McDonnell, the shadow chancellor.

If the analysis is correct, the political significance is enormous. Jeremy Corbyn and his allies are currently reluctant to speak up for a People’s Vote and the chance to stop Brexit. This is partly because they fear the response in Leave-voting Labour heartlands. If these are now moving sharply away from wanting Brexit, then the danger of a backlash is clearly reduced.

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But is the analysis correct? Large as it is, a poll of 15,000 means that YouGov gathered data from an average of just 24 people per constituency. However, YouGov faced similar criticisms when it predicted last year that Conservative strongholds such as Canterbury and Kensington might fall to Labour. They duly did. The reason why YouGov was right was not that it polled hundreds of people in both places – it didn’t – but because it built up a detailed picture of party support among different demographic groups, and then applied this data to the specific characteristics of each constituency.

In the latest exercise, YouGov’s data has been analysed by consumer analytics firm Focaldata, using broadly the same approach. While there is plainly a margin of error in the figures for individual constituencies, I believe the overall picture is about right. For one thing, YouGov surveys this year have invariably found that far more Labour Leave voters are now having second thoughts than Conservative Leave voters. We should, therefore, not be surprised that Labour-Leave constituencies are moving more towards Remain than Conservative-Leave constituencies.

Both recent big YouGov polls report that a referendum held now would produce a 53-47% victory for Remain. This is a 5% swing from the 2016 result of 52-48% for Leave. It follows that the swing in many Labour-Leave constituencies will be significantly greater than 5% – which is precisely what Focaldata are saying.

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The second YouGov poll was conducted this month for the People’s Vote campaign among a sample of more than 10,000. It tells us eight key things about why the public is on the move

As many as 73% of voters, including 60% of Leave voters, now agree that “it is likely that many of the promises made by politicians in favour of leaving the EU will be broken”. Just 13% disagree.

By almost two-to-one (43-22%), voters fear that Brexit would harm rather than boost Britain’s economy.

Few voters expect Brexit would enable the UK to “take back control”. Just 21% believe the UK would in practice gain the freedom to decide its own rules and regulations; 51% think we “will have to obey many EU rules and regulations if British businesses are to continue to trade freely with other European countries”.

Free and frictionless trade is seen as vital. If forced to choose, 50% would opt for free trade rather than the right to impose immigration controls; just 29% give immigration controls a higher priority.

If Brexit goes ahead, fully 70% want EU citizens to continue to have either an absolute right to settle in the UK (16%) or freedom for workers and students to come to the UK (54%). Thus freedom of movement of labour is popular, even if untrammelled freedom of movement for people is not. As other EU countries in practice qualify freedom of movement by drawing precisely that distinction, EU immigration need not be a stumbling block to a new deal that enables the UK to remain a member of the EU.

Given these findings, it makes sense that, by 45-34%, voters now favour “a public vote on the outcome of the negotiations”.

Furthermore, YouGov went on to ask a completely new question: what should happen if talks break down and there is no deal between London and Brussels: “should the final decision be made by MPs voting in Parliament or the public voting in a new referendum?” 50% want a public vote, while just 25% opt for MPs voting in Parliament.

Even people who voted Leave tend to prefer a new referendum, by 39-34%.

There is clearly the potential for a broadly-based campaign this autumn for a People’s Vote, should the Brussels talks go badly. Across the spectrum, the message from voters is clear: if the government and Parliament can’t sort out Brexit, the people should.

This article was originally published with the incorrect summary on the InFacts home page and in the InFacts daily newsletter. This has now been rectified.

Edited by Luke Lythgoe