Voters would back remaining in the European Union by 53% to 47% if a referendum was held now, according to a major poll.
Some 45% want a say on the outcome of the Brexit negotiations with 34% opposed, the survey for the People's Vote campaign found.
If talks break down, half of voters said the final decision over whether the UK should crash out of the bloc or remain in the EU should be taken by the public and a quarter said Parliament should decide.
Opponents of Brexit said the YouGov study of more than 10,000 adults showed public opinion was shifting.
Peter Kellner, a leading pollster and former president of YouGov, said the findings were "politically significant".
He said: "Across the spectrum, the message from voters in this survey is clear: if the Government and Parliament can't sort out Brexit, the people should. "
Labour voters overwhelmingly said they would back remaining in the EU, some 77% to 23% for Leave.
Most back a referendum on the exit deal, 63% compared to 18%.
Mr Kellner said the polling showed the chasm between Labour members and the leadership over its approach to Brexit.
"Labour supporters overwhelmingly back a new public vote and, in resisting this, the party's leadership is increasingly out of touch with its own voters," he said.
"But this exceptionally large 10,000 sample survey contains more than 2,700 current Labour supporters, leaving no room for doubt about the size of the majority among them for a new public vote."
The 53% to 47% result in favour of remaining in the EU excludes those who do not know how they would vote or would not bother.
A no deal scenario pushes the six-point margin to 12%, 56% Remain to 44% Leave, the study found.
Nearly three quarters, 73%, believe many of the promises made by the Leave campaign will be broken and 68% fear the UK will end up with a bad deal, with 64% ready to lay the blame on the Government.
Labour's Chuka Umunna, who backs the People's Vote campaign, said: "Politicians of all sides are increasingly trapped inside red lines of their own making, but the momentum this summer is with the campaign for a people's vote as more and more people realise it's the only way to take back control of the process.
It is clear from YouGov's findings that public opinion is moving behind a people's vote across the political spectrum. (3)— Chuka Umunna (@ChukaUmunna) August 10, 2018
"It is clear from YouGov's findings that public opinion is moving behind a people's vote across the political spectrum.
"This poll shows that Labour voters overwhelmingly support a people's vote, putting them at odds with the party's official pro-Brexit position.
"The Labour Party must now do what its members and supporters and voters are crying out for: put clear red water between the Opposition and the Government, and provide leadership to the country by backing the People's Vote campaign."
10,121 adults were surveyed online between July 31 and August 7. The figures have been weighted and are representative of all GB adults.