David Cameron could slash Ukip's support by more than a third if he promises an in-out referendum on EU membership, according to a poll.
Research by ComRes for the Sunday People found 63% of the public want a vote on whether Britain should remain in the union.
However, more people - 42% said they were against leaving the EU.
There were also signs that opposition to the union has softened, with the proportion who think there should be a referendum dropping from 68% in October 2011.
At that time 37% wanted to exit the EU altogether.
The poll suggested that Ukip is on track to knock the Tories into third place in next year's European parliament elections.
Asked who they would back in the contest, 35% said Labour, 23% Ukip and 22% the Conservatives. The Lib Dems were on just 8%.
But as the Prime Minister prepares to make his crunch speech on Europe later this month, Ukip voters were asked how they would react if he pledges to hold an in-out referendum.
Nearly four in 10 - 37% - said they would probably not support the party any more.
ComRes chairman Andrew Hawkins said: "While European and Westminster electoral dynamics are different, the prospect of humiliation in 2014 would fuel disquiet among Mr Cameron's right flank who have still not forgiven him for not winning in 2010 and want to see traditional Tory values asserted more aggressively.
"The challenge for (Ukip leader) Nigel Farage is to appeal now to Labour voters and to maintain his party's support in the event of the promise of a referendum."
Earlier on Saturday, Tory grandee Lord Heseltine turned on David Cameron, warning the PM against taking a “punt” by holding a referendum on Britain’s continued membership of the European Union.
The former minister, who served under Margaret Thatcher, made the comments in interviews with The Times and FT ahead of Cameron’s speech on the EU later this month.
Cameron has said he wants the UK to stay in the EU, but wants to renegotiate the terms of the relationship, particularly as those within the EU are pushing for greater integration.
However the PM's desire to fashion a looser relationship with Europe could prove tricky. On Thursday, a delegation of German MPs told the prime minister not to "blackmail" the rest of Europe with threats, while on Wednesday the Obama administration warned Britain not to turn "inwards" with a referendum.
ComRes interviewed 2,059 adults online between December 19 and 21. Data were weighted to be representative of all adults. The Ukip voter sample size was 304.