When the survey’s findings are projected on to last week’s vote, they would cut the Leave share by 1.2 million, almost wiping out the majority that gave Friday's shock result.
Research by Opinium found that 3 per cent of those who voted Remain also regretted their choice and that British people are now divided on the priorities in the negotiations ahead.
More than half of those surveyed also felt both the UK’s economy and position in the world had worsened, but almost one in 10 said they did not believe the Brexit would be implemented.
The findings came after some voters expressed “Bregret” over their choice following the immediate impact markets and U-turns on Leave campaign pledges including the promise to spend £350 million a week on the NHS.
“I wish we had the opportunity to vote again,” Leave voter Mandy Suthi said, adding that she was “very disappointed”.
“I personally voted leave believing these lies, and I regret it more than anything, I feel genuinely robbed of my vote,” wrote Khembe Gibbons, a lifeguard from Bury St Edmunds in Suffolk,
A voter who gave his name as Adam told the BBC he would have changed his pro-Brexit vote if he knew the short-term consequences it would have for the UK economy.
"I'm shocked that we voted for Leave, I didn't think that was going to happen,” he said. “I didn't think my vote was going to matter too much because I thought we were just going to remain."
More than 4 million people have signed a petition calling for a second EU referendum but the Government has ruled out another vote on the issue, dubbed a “neverendum” by David Cameron.
Brexit reactions – in pictures
Brexit reactions – in pictures
Supporters of the Stronger In campaign look at their phones after hearing results in the EU referendum at London's Royal Festival Hall
Leave supporters cheer results at a Leave.eu party after polling stations closed in the Referendum on the European Union in London
Mr Cameron announces his resignation to supporters
Donald Tusk proposes that the 27 remaining EU member states ‘start a wider reflection on the future of our union’
Ukip leader Nigel Farage greets his supporters on College Green in Westminster, after Britain voted to leave the European Union
Supporters of the Stronger In Campaign react as referendum results are announced today
Boris Johnson leaves his home today to discover a crowd of waiting journalists and police officers
Leave EU supporters celebrate as they watch the British EU Referendum results being televised at Millbank Tower in London
Supporters of the Stronger In Campaign react as results of the EU referendum are announced at the Royal Festival Hall
Supporters of the Stronger In campaign react after hearing results in the EU referendum at London's Royal Festival Hall
Adam Drummond, from Opinium Research, said the firm’s research showed deep divisions over the vote for a Brexit and how the UK should move forward.
“The UK is just as divided post-referendum as it was pre-referendum with voters split on what the UK’s relationship with the EU should be after we leave and what the priority should be in the ensuing negotiations,” he added.
“Remain voters want the government to prioritise staying part of the EU’s single market while Leave voters are keen to end free movement between the UK and the EU and both priorities are likely to be mutually exclusive.”
Around 43 per cent of people thought the UK was unlikely to be able to remain in the single market, which more than a third of respondents said was their utmost priority.
Other key issues were curbing immigration and ending the free market of labour.
The majority of those questioned also wanted a general election to be held before official negotiations on the Brexit begins, with Theresa May the favoured candidate to take David Caomeron’s place as Prime Minister among Conservatives and Jeremy Corbyn among Labour.
Opinion polls in the months leading up to last week’s historic vote had dominantly shown a lead for Remain, although surveys in recent days showed the result on a knife-edge and around 10 per cent of the electorate still undecided – generating a huge swing.
The final result was 17,410,742 votes for Leave (51.9 per cent) compared to 16,141,241 for Remain (48.1 per cent), on a turnout of 72 per cent.