According to research carried out for The Times, 40% of voters back an UK exit while 37% want to keep ties with Brussels and 23% do not know.
Britain's future in the European Union would rest on a knife-edge if a referendum was held immediately, polling has indicated.
The Populus survey would translate to a 53-47 vote in favour of leaving after other factors, such as likelihood to vote, were taken into account, according to the newspaper.
Prime Minister David Cameron has insisted he wants Britain to remain within the EU, although under a renegotiated settlement. Meeting fellow EU leaders for the first time since announcing his plan to stage an in/out referendum on UK membership, he insisted the British people were not "turning our backs on Europe".
But he told them Europe was being "out-competed, out-invested, out-innovated" and must be transformed into an engine for growth.
Attending the World Economic Forum in Davos, Switzerland, Mr Cameron discussed the proposals in a 15-minute meeting with German chancellor Angela Merkel, who has indicated she is open to a "fair compromise" with Britain.
Deputy Prime Minister Nick Clegg said he is "still none the wiser about exactly what this great re-negotiation means".
In an interview with The House magazine, he said: "It was a well-crafted speech and obviously very well delivered. But in terms of the content, look, the Conservative Party will obviously ride away with headlines about the referendum today. My own view is that it will be a tactical victory today for a strategic mistake tomorrow."
Meanwhile, Labour leader Ed Miliband sought to clarify his own position on Europe, after appearing to rule out an in/out referendum in the House of Commons. He said: "I am being clear. I do not think it makes sense, now, to commit to an in/out referendum years ahead.
"And the reason why it does not make sense is clear from what the priority of the British people is. Their priority is jobs and growth and living standards and I've got to say what I think the right priority is and I do not believe now it makes sense to commit to an in/out referendum."