The poll, published by the pro-European GMB union, will send a strong signal to ministers to "get off the fence" on the euro and will increase pressure on them to take a more positive line in the run-up to the next general election.
The debate over Europe will reignite at the conference in Bournemouth this week with several prominent Labour figures, including Peter Mandelson and Robin Cook, the Foreign Secretary, making pro-European speeches which highlight the potential benefits to business of joining the euro.
Ministers have come under fire for failing to provide positive leadership in the European debate. They were criticised for avoiding discussion of the euro in this year's European elections which saw Labour lose seats to the Conservative party.
But next month Robin Cook will warn that Britain's influence in Europe could be jeopardised by not joining the euro. He is to lead the pro-single currency debate within government which believes that the decision to join should be based on economics.
The Foreign Secretary is the first senior Cabinet minister to warn of the political dangers of ruling out entry into the single currency. His remarks, and the results of the poll, will delight pro-Europeans within the party.
The poll shows that the gap between those opposed to the single currency, who are still in the majority, and those willing to vote yes in a referendum would close sharply if the Government were to adopt a more positive line on the issue.
The Mori poll shows that the gap between those opposed to abandoning the pound and those in favour of joining the single currency has fallen from -31 points in July to -22 today.
But when respondents were asked how they would vote, if the Government were to urge strongly that Britain should be part of a single European currency, the gap narrows further to just -12.
"This poll shows that the supertanker of British public opinion is gradually starting to turn," said John Edmonds, general secretary of the GMB.
"But what it also reveals is the extent to which success or failure in a referendum campaign lies squarely in the hands of the Prime Minister. If Tony Blair and his ministers retain the courage of their convictions and go for it, then it's clear that the British people are ready to be won over."
The Britain in Europe Campaign, which will shortly be launched with the support of Tony Blair, Charles Kennedy, Kenneth Clarke and Michael Heseltine, is to host a series of high-profile debates on Europe at this week's Labour conference.
"Europe is a key theme at the conference," said a spokesman for Britain in Europe. "We want to retain the option to join the single currency in the next parliament."