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An ITV News/ComRes poll has found that the proportion of Britons that would vote for the UK to stay in the EU has fallen by four percentage points.
The figure has dropped to 36 per cent following the row over Jean-Claude Juncker’s appointment as President of the European Council last week.
If a referendum was to be held, 43 per cent of people would vote for Britain to leave the EU.
And 34 per cent believe membership of the EU is a “good” thing for the country, a fall of three per cent, while 44 per cent say it is a “bad” thing.
David Cameron is set to be grilled by MPs in the Commons later over his failed bid to stop Jean-Claude Juncker being chosen as the next president of the European Commission.
The Prime Minister has faced criticism of his negotiating tactics after he proved unable to block Mr Juncker's nomination to the crucial post on Friday.
Tories have insisted he is set to receive "amazing support" from his backbenchers over the position he took.
However, he is likely to get a rougher ride in the Commons from Labour, whose shadow chancellor Ed Balls has branded the failed negotiations a "catastrophe" for Britain.
Writing in The Daily Telegraph today, Cameron has insisted he can still secure reforms to Britain's relationship with Europe, saying: "I am ready to move on and keep fighting for Britain's interests in Europe."
David Cameron has admitted that his opposition to Jean-Claude Juncker becoming President of the European Commission has made renegotiating the UK's position in Europe more difficult.
"I do not deny that it has made the task harder and the stakes higher," the Prime Minister writes in today's Daily Telegraph.
However he insisted the Government would not be deterred from trying to reform the EU.
"When we encounter setbacks, we don’t throw in the towel: we redouble our resolve. The task of reforming Europe and securing Britain’s place in a reformed Europe was always going to be a long and tough campaign," Mr Cameron argues.
"You don’t turn around a tanker like the EU with ease; this will be tough, and we’ve always known that," he added.
David Cameron has insisted he can "do business" with the new President of the European Commission, despite having vigorously opposed his appointment.
Writing in the Daily Telegraph, the Prime Minister admitted he was "isolated" over the decision to instal Jean-Claude Juncker at the head of the EU's executive branch, but insisted he was "right" to take a strong stance on the issue.
At the same time, Mr Cameron stressed he was now "ready to move on and keep fighting for Britain’s interests in Europe".
He remained clear that his government was strongly opposed to being involved in further EU integration, saying: "If...we can agree that we are not heading, at different speeds, to the same place – as some have assumed up to now – then there is business we can do".
Despite campaigning against him, David Cameron called Jean-Claude Juncker today to congratulate him on securing the nomination to be the next president of the European Commission, Downing Street said.
A spokesman for Number 10 said: "The Prime Minister called the Commission President-designate, Jean-Claude Juncker, this afternoon. The Prime Minister congratulated Mr Juncker on running a successful campaign and securing the Council nomination."
They added: "They discussed how they would work together to make the EU more competitive and more flexible. The PM welcomed Mr Juncker's commitment of finding a fair deal for Britain and Mr Juncker said that he was fully committed to finding solutions for the political concerns of the UK."
Juncker, the former Luxembourg premier told the Prime Minister he was "fully committed" to finding a solution to British concerns about the European Union and they discussed "how they would work together" to boost competitiveness, they said.
The EU without the UK is "unimaginable" Germany's finance minister Wolfgang Schaeuble said in an interview with the (£) Financial Times.
He said: "The EU without the UK is absolutely not acceptable, unimaginable. Therefore we have to do everything, so that the interests and the positions of the UK find themselves sufficiently in European politics."
Germany will do everything its power to the UK within the European Union, according to the country's finance minister.
Speaking to the (£) Financial Times, Wolfgang Schaeuble said: "Historically, politically, democratically, culturally, Great Britain is entirely indispensable for Europe."
He added that Britain's exit from the union was "unimaginable."
Shadow chancellor Ed Balls said the Prime Minister's actions in Brussels this week were "catastrophe for Britain and the British national interest."
"We won't be influential in the world, unless we are influential in Europe," he told BBC One's The Andrew Marr Show.
"I've never seen a negotiation so cack-handed," Mr Balls said about David Cameron's failure to stop the appointment of Jean-Claude Juncker as the new head of the European Commission.
Mr Balls said about the Prime Minister: "He's weak, he's lost control, he's on the back foot, Britain is suffering. I think it's catastrophic for Britain."
David Cameron is following in the footsteps of Harold Wilson on Europe, former Conservative Chancellor Lord Nigel Lawson has said, after the Prime Minister failed to halt the appointment of Jean-Claude Juncker to the European Union's top job.
Speaking to the Independent on Sunday, Lord Lawson said: "He might do a Harold Wilson – whether the public will believe it as they did in 1975 is another matter [...] he was going to renegotiate the terms and then put it to the people in a referendum".
"Through the long renegotiation, we got absolutely damn all [...] but he presented it as a great success, and people bought it. I think David Cameron could try to do the same; he will get very little and he will present it as something", he said.
He added: "If David Cameron had said, 'Well I'm not sure how to vote, it depends on what I am able to negotiate' – that would have been a stronger hand. But he's actually made it clear he's going to vote for 'in' irrespective, so he has no negotiating hand to speak of."
The leader of Britain's biggest business group has said that the country's economic success depends on it remaining a full member of the EU, after senior Tories revealed that more than 150 of the party's MPs would campaign to leave the union in a referendum.
CBI director general John Cridland told the Observer that full membership of the EU boosted British jobs, growth and investment.
"The EU is our biggest export market and remains fundamental to our economic future," he said. "Our membership supports jobs, drives growth and boosts our international competitiveness."
Latest ITV News reports
It should be no surprise that David Cameron failed to stop such a long-standing EU figure being nominated as European Commission president.
The PM's friends in Europe are bewildered that he would rather go down in flames than try for the best deal possible in the circumstances.