Proposals for reforming the European Union have failed to win over Britons with 45% intending to vote for the UK to leave, according to polling.
Support for Brexit has increased in the face of a package of measures that would change the UK's relationship with Brussels, research for The Times found.
The poll was carried out in the two days after publication of the outline deal and found the number of voters wanting to quit had risen by three points on the previous week.
Some 36% of voters want Britain to remain in the 28-strong bloc but nearly a fifth, 19%, are yet to decide or plan not to vote, the YouGov survey said.
It is the latest blow for Mr Cameron, who is fighting for support on a number of fronts.
The Prime Minister has faced continued criticism of the deal from within his own party and Cabinet minister John Whittingdale has indicated he stands ready to campaign for Britain to leave.
Mr Cameron is embarking on a fresh diplomatic dash round Europe to push for support from counterparts in Poland and Denmark ahead of the crunch meeting on the proposals later this month.
It follows a series of talks with leaders, including European Council president Donald Tusk, German chancellor Angela Merkel and Polish PM Beata Szydlo, in the margins of the conference on the Syrian crisis in London
Mr Cameron is holding further talks with Ms Szydlo in Warsaw before heading to Copenhagen for a meeting with prime minister Lars Lokke Rasmussen to seek support for reform.
The Prime Minister has agreed to meet the heads of the political groups in the European Parliament on February 16, two days before the leaders' summit.
But Mr Cameron is facing a tough task at home as Eurosceptics in his party continue to heap criticism on the package.
Mr Whittingdale has become the most senior figure in the party to suggest he is ready to campaign for Britain to quit since the proposals were put on the table by Mr Tusk on Tuesday.
The Culture Secretary pointed to his track record of being "highly critical" of the way the 28-strong bloc is run as he refused to rule out backing a Brexit.
He told The House magazine: "My position is that the Prime Minister is out there trying to negotiate the best deal he can for the country.
"I have a track record where I've been highly critical of the way the EU works and I have opposed measures for closer integration and it certainly needs reform.
"I hope the Prime Minister will get that agreement and then I'll look at it when he comes back with it."
Asked if he would rule out backing Brexit, he replied: "I wouldn't."