First UK poll after Juncker battle shows rising support for EU exit

* Poll shows more Britons want to leave the EU than stay

* Other polls this year have shown falling support for exit

By William James

LONDON, June 29 (Reuters) - Public support for Britain to leave the EU is growing again after falling to a multi-year low, according to the first survey published since EU leaders nominated Jean-Claude Juncker to become the next president of the European Commission.

British Prime Minister David Cameron openly campaigned to stop Juncker getting the top EU job, but last Friday lost the final vote by 26-2. He warned at the time that choosing Juncker, an old-style EU federalist, would make it harder to keep Britain in the European Union.

Under pressure from Eurosceptic lawmakers in his own Conservative party and from the anti-EU UK Independence Party (UKIP) which won European parliamentary elections last month, Cameron has pledged to try to reshape Britain's EU ties before giving people a referendum in 2017 on whether the UK should remain a member of the EU.

He can only deliver a referendum if re-elected next year.

Many though not all opinion polls conducted this year have shown public support to leave the EU falling back. But Sunday's poll published by the Mail on Sunday newspaper, the first since Juncker's nomination, showed 47 percent would vote for Britain to leave the bloc if there was a referendum, with 39 percent saying they would back staying in.

The polling for the Mail on Sunday by Survation, which was conducted among 1,000 adults on Friday, the same day as Cameron lost the Juncker vote, also showed that 30 percent of people felt Juncker's nomination made it more likely they would vote to leave the bloc if given the chance, while only 13 percent of Britons said they backed Juncker for the job.

The poll contrasted with one conducted by YouGov (Other OTC: YUGVF - news) earlier this month which showed support for Britain staying in the EU at 44 percent, its highest level in four years. It showed only 36 percent wanted to leave.

Sunday's poll was published a day after one of Cameron's senior ministers raised the prospect that unless Europe pressed on with reforms, the Conservative party could campaign for an exit in a national referendum on EU membership.

Cameron has previously always said he wants to be able to campaign for Britain to remain inside a reformed EU.

"We are not going to recommend a deal to the British people unless it's the right deal," Jeremy Hunt, Health Secretary and a Cameron ally said on BBC radio.

The opposition Labour party, which is refusing to match Cameron's promise of an EU referendum, said his Juncker strategy was "cack-handed" and risked pushing Britain closer to the EU exit door. UKIP has said Cameron picked the wrong EU fight and as a result weakened Britain's clout in the EU.

Europe minister David Lidington said on Sunday that Britain had to put pressure on Juncker to pursue reforms, citing European election results last month which showed a rise in support for eurosceptic parties in several EU countries.

"I don't think he will want his commission to be marked by the exit of one of the biggest, most important member states," Lidington said of Juncker. (Editing by Andrew Osborn and Greg Mahlich)