David Cameron has been accused by the European commission president of having a “problem with other prime ministers”, in an escalation of tensions between the UK and Brussels.

Jean-Claude Juncker, whose appointment was strongly opposed by Cameron, publicly swiped at the UK for refusing to pay its extra £1.7bn contribution towards the EU budget.

In his first press conference since taking the job, Juncker struck a defiant attitude to the UK’s rejection of the bill, saying he was “not the type who trembles, in front of prime ministers or at any other time” and adding that he was not frightened of any of them.

“I don’t have a particular problem with Mr Renzi [the Italian prime minister], whom I have great respect for. I don’t have a problem with Mr Cameron; Mr Cameron has a problem with the other prime ministers,” he said.

Juncker’s suggestion that Cameron is isolated among his fellow leaders follows the UK’s calls for its contribution to the EU budget to be re-examined. All countries have to pay adjusted sums, but no others are as high as the UK’s extra £1.7bn, which must be paid by 1 December or start accruing interest.

However, there are signs that the EU could be willing to compromise, with one official telling Reuters that one option would be allow Britain to pay in instalments.

Downing Street dismissed Juncker’s comments on Wednesday, with the prime minister’s spokesman saying Cameron works well with other European leaders. He also would not comment directly on the idea of paying in tranches, simply saying Cameron objects to the scale of the demand.

The £1.7bn bill has put Cameron under intense pressure at a time when the Conservatives are fighting to keep the constituency of Rochester after their MP Mark Reckless defected to the anti-EU Ukip.

Earlier on Wednesday, Cameron clashed with the Labour leader, Ed Miliband, over his refusal to say if he would ever campaign for Britain to leave the EU if he was unable to secure his objectives in a renegotiation.

Pressed by Miliband at prime minister’s questions, Cameron said repeatedly: “I could not be more clear I want Britain to stay in a reformed European Union.”

The Labour leader accused Cameron of being torn between the demands of Conservative backbenchers, who want to leave the EU, and the national interest that requires Britain to remain.

Miliband also quoted from a 2010 Daily Telegraph interview in which Cameron said he would “not countenance leaving the EU and … would never campaign for an out vote in an EU referendum”.

Cameron has for months been reluctant to say what he would do if he fails to secure his negotiating objectives, an increasingly likely possibility after aides to the German chancellor, Angela Merkel, made clear that she would not abandon the principle of the free movement of EU citizens within the EU. She has said she is willing to look at the restrictions of benefits.

The prime minister has not specified whether he was seeking to restrict free movement or reduce the so-called “pull factor” by restricting access to benefits including in working tax credits. He is due to set out detailed plans for renegotiation in a speech before Christmas.

Pressed by Miliband on whether he had support from any of his EU partners for his renegotiation package, Cameron said: “What we have is a set of things that we want to sort out in Europe. We want to sort out safeguards for the single market. We want to get out of ever-closer union. We want reform of immigration. But here’s the difference: we have a plan; he has no plan. And we have a plan which will be put to the British people in an in-out referendum.”

He also insisted that Germany was sympathetic to his demands, saying: “[Merkel] has herself said there are problems in terms of free movement that need to be dealt with”.

However, with a CBI conference coming up next week, Miliband will be hoping that business will start to voice its concerns that Cameron, scared by the progress of Ukip, is gradually raising its negotiation demands so much that he will end up in a position of leaving the EU. Conservatives claim business is increasingly reconciled to the need for a referendum.

Cameron spent most of his time in his answers attacking Miliband’s leadership. He said: “The Labour party right now is in a dreadful position.” Quoting the admission from the deputy chief whip, Thomas Docherty, that Labour was a moribund party in Scotland, Cameron crowed: “It’s official: it’s a dead parrot.”

He also quoted the former Labour chancellor, Alistair Darling, who had claimed the boil of a referendum on Europe had to be lanced.