The survey, conducted by pollster Opinium Research for pro-EU campaign group Open Britain, saw 31% of respondents say they were opposed to a "people's vote", while 17% said they didn't know or had no opinion.
May has repeatedly said she will deliver what she calls the will of the people and on Sunday, Foreign Secretary Boris Johnson voiced optimism for the kind of deal Britain could win.
Actor Patrick Stewart, best known from Star Trek, is also backing the campaign.
The campaign is run by Open Britain, which is one of several anti-Brexit organisations which sprung up during the 2016 referendum.
"Brexit is not inevitable".
"Now that we are learning the real cost of Brexit, I want to urge that we think again and insist whether we accept the Brexit deal is a matter for the people", he added. "It should not be a done deal".
"Brexit will affect everybody in the country, which is why it should not be left to 650 politicians to decide our future but 65 million people. Together, are on the verge of something extraordinary".
The British parliament's only Green MP, Caroline Lucas, said: "This is too big and too important to be determined exclusively by politicians. If the public demand a People's Vote, politicians will fall in line".
The bloc has previously stated the United Kingdom can not stay in the Single Market after Brexit and end open borders, and Mr. Barnier said Prime Minister Theresa May would have to drop her "red lines" on migration to do so.
He said the group does not oppose the result of the 2016 European Union referendum or call for a second referendum. The terms of the separation have not yet been finalised.
Some opinion polls have shown a slight shift in favour of remaining in the European Union, but there has yet to be a decisive change in attitudes and many in Britain say they have become increasingly bored by Brexit.
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