The last time Britain held a vote on staying in Europe was in a 1975 referendum when the "Yes" campaign came from behind to win by a wide margin, with 67.2 percent in favor of European membership and 32.8 percent against.
That result failed to take the sting out of an issue that has divided the main British parties for years and helped to bring down two of Cameron´s predecessors, Margaret Thatcher and John Major.
Cameron said in January he would campaign for Britain to remain in the EU "with all my heart and soul", as long as he repatriates powers. That would set him against "Eurosceptics" in his Conservatives and the many voters who see the EU as a threat to their sovereignty.
Twelve percent of those planning to vote to leave the EU said they would definitely change their minds if Cameron clawed back powers. Another 47 percent would "possibly" rethink their opposition.
As in 1975, the "Yes" campaign would be well funded by business leaders warning of the economic dangers of Britain walking away from a single market of 500 million people.
Cameron would also have the support of the pro-EU Liberal Democrats, his junior coalition government partner, and the main opposition Labour Party, which supports Britain´s membership.
Labour leader Ed Miliband, ahead of Cameron in the polls, visits Denmark, Sweden and the Netherlands this week to discuss Britain´s EU ties.
While British politicians have long had a feverish debate about Europe, the poll said voters do not share their obsession. Europe came 14th in a list of 15 priorities, far behind health, education and the economy.