The prime minister has said "significant progress over the coming days" would be needed
More than half of voters back the UK staying in the European Union, according to the latest opinion poll.
The lead for the 'remain' camp over those wanting to 'leave' came as David Cameron was set to continue a frantic round of talks in a bid to secure changes in Brussels ahead of an in-out referendum.
The British prime minister rejected as "not good enough" a proposal to create an emergency brake that would allow EU countries to stop paying in-work benefits if their public services are overwhelmed.
Cameron's office has also warned "significant progress over the coming days" would be needed.
The negotiations are entering a critical phase in the run up to a crucial summit next month.
European Council president Donald Tusk is due to attend a dinner at Number 10 Downing Street in London on Sunday to finalise "concrete" proposals which are expected to be published on Monday.
In a phone call with the French President Francois Hollande, a spokesperson for Mr Cameron said "a deal in February remained possible if significant progress can be made in the coming days".
A spokeswoman added: "The Prime Minister said that if the deal on the table for February wasn't right, he wouldn't take it - there is no hurry and it is more important to fundamentally reform the UK's relationship with the EU."
UKIP leader Nigel Farage accused Mr Cameron of trying to "fiddle" with his pledge to tackle EU migration.
He said: "He's fiddling around the edges, begging for concessions that we may get on granting migrants benefits, and given the first offering he's had from Mr Juncker, he will now go to the European summit in February and resemble the boy from Oliver! who walks up to the front of the room and holds out the begging bowl and says 'please sir, can we have more concessions'."
Meanwhile, a ComRes opinion poll for the Daily Mail found voters in favour of the UK staying in the EU by 54% to 36%, with 10% undecided.
ComRes interviewed 1,006 adults by telephone between 22 and 24 January.