Labour could win more than 1.5 million extra voters if it backed a second referendum on Brexit, potentially delivering the party a Commons majority, according to new polling for the campaign group People’s Vote.

Labour’s policy on Brexit will be a central issue at its conference in Liverpool, which kicks off on Saturday, with more than 100 local parties urging the leadership to back a referendum on any final Brexit deal.

The poll is aimed at allaying the anxieties of Labour voters in pro-leave constituencies, who fear that opening up the possibility of revisiting the Brexit vote could alienate their supporters.

While the organisers stress the complexities of asking such hypothetical questions, they argue that the large sample size – more than 10,200 people were asked in late August and early September – plus the heavy margins support the idea of an electoral boost if Labour changes policy.

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The survey, carried out by YouGov, found that 26% of people said they would be more likely to vote Labour if it campaigned for a “people’s vote”, against 6% who said they would be less likely.

YouGov calculated this to mean such a policy change would see the party gain just over 1.5m votes overall, with an extra 1.7 million people supporting the party and around 200,000 withdrawing their support.

If this new support was evenly distributed across constituencies – which is very unlikely – the extra 2,400 votes per area would win Labour about 66 more seats in all, just enough for a majority if other seats were not lost, based on the 2017 election.

The former YouGov president Peter Kellner, who analysed the data for People’s Vote, said that while some Labour MPs argue backing a second referendum could cost the party net votes, the poll figures do not bear this out.

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He said: “Of the respondents who currently support Labour and voted leave in the Brexit referendum, only 6% put Brexit at the top of their concerns and say they would not, or be less likely to, vote Labour if the party backed a popular vote. This is just 2% of all Labour supporters, or just over 200,000 voters in all.

“In other words, the non-Labour voters that the party could win over outnumber the Labour voters that the party risks losing by almost nine to one.”

Labour’s official policy on a second Brexit referendum is to not completely rule it out, but to argue strongly that a better option would be to hold a general election, allowing the chance for a Jeremy Corbyn-led government to negotiate a departure.

The poll also found 55% to 45% backing for a second referendum, with 50% of those asked identifying Brexit as the biggest issue facing parliament. Perhaps more expected was the 84% of respondents who described the Brexit negotiations so far as a mess.

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People’s Vote is holding a march in Liverpool on Sunday to coincide with the Labour conference, titled the “People’s Vote March for the Many”.

David Lammy, the Labour MP for Tottenham, who will be one of the speakers, said a second referendum was “the right thing for the party to do if it is to stand up for the communities, public services and young people who will be hit first and worst by the kind of Brexit that people like Michael Gove, Jacob Rees-Mogg and Boris Johnson want to impose on us”.

He added: “But what these figures show is that supporting a people’s vote is also the right thing for the party to do if we it wants to win a general election. This should concentrate some minds at our conference in Liverpool next week.”