On Thursday, Cameron will present the European Council with his case for EU reforms. The prime minister has repeatedly said that the United Kingdom would remain in the bloc, but only if the European Union implemented reforms.
"Failure to secure agreement on key objectives significantly increases the prospect of a Leave vote," New Open Europe/ComRess poll said.
The poll results suggest that if the referendum were held tomorrow, 56 percent of voters would vote to remain in the European Union, and 35 would vote to leave.
However, according to the poll, if Cameron achieved his demands, 65 percent of voters said they would vote to remain in the bloc.
London is looking to curb social benefits for EU migrants, for more powers to be granted to the bloc member’s national parliaments, greater protections for non-euro countries in the bloc, measures to strengthen the European single market, and to ensure that the United Kingdom is not obliged to adhere to the EU’s commitment to an "ever closer union."
ComRess interviewed 1,001 British adults via telephone on December 11-13.