A defiant Theresa May has insisted her efforts to win fresh assurances on her Brexit deal remain on track after another bruising encounter with EU leaders.

The Prime Minister said further talks would take place in the coming days on measures she hopes will persuade MPs to back the agreement in Parliament.

EU leaders acknowledged the need to “bring down the temperature” after appearing to rebuff her calls for assurances on the Northern Ireland backstop at a late-night press conference in Brussels.

At the same time however, European Council president Donald Tusk said he had “no mandate” to open new negotiations, although he would remain at Mrs May’s “disposal”.

Earlier the tensions erupted into public view when a visibly angry Mrs May was seen berating European Commission president Jean-Claude Juncker over remarks he had made during the press conference in the early hours of Friday.

The Prime Minister was apparently infuriated after he described the British negotiating position as “nebulous and imprecise”, and called on the Government to spell out exactly what it wanted from the talks.

Mr Juncker said he had been able to convince her that his remarks had referred to the wider Brexit debate in the UK, joking: “After having checked what I said yesterday night, she was kissing me.”

However there was no disguising the disappointment on the British side that Mrs May came away from the two-day summit in the Belgium capital with only limited assurances regarding the backstop, which is intended ensure there is no return of a hard border on the island of Ireland.

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The Irish border which has proven a thorny issue in Brexit talks (Michael McHugh/PA)

The EU agreed they should try to ensure the backstop was never needed by swiftly working to reach a new trade agreement with the UK after it has left, and to find alternative arrangements as soon as possible if it was activated.

However, at the end of a bruising week, in which Mrs May survived a vote of no confidence by Tory MPs there were hopes there would be some move towards providing a legal assurance that the UK could not trapped in the backstop indefinitely.

It followed a direct appeal by Mrs May to the other 27 leaders to help her to get the deal through Parliament where she is facing opposition from all sides including many of her own MPs and the DUP who prop up her Government in Parliament.

In her end-of-summit press conference, she acknowledged that the Withdrawal Agreement could not be reopened, but insisted their discussions had shown there was scope for “further clarification” on the way the backstop would operate.

“The EU is clear, as I am, that if we are going to leave with a deal, this is it,” she said.

“But my discussions with colleagues today have shown that further clarification and discussion following the council’s conclusions is in fact possible.

“There is work still to do and we will be holding talks in coming days about how to obtain the further assurances that the UK Parliament needs in order to be able to approve the deal.”

On the EU side there was exasperation at the inability of MPs at Westminster to agree on a way forward amid concerns that further concessions on the backstop would still not be enough for Mrs May to get the deal through the Commons.

Mr Juncker said: “Because I was following second by second the debate in the House of Commons and I noted there was a deep mistrust in the House when it comes to the European Union.

“That is not a good basis for future relations.

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European Commission president Jean-Claude Juncker embraces council president Donald Tusk at the Brussels summit (Stephanie Lecocq/AP)

“We have to bring down the temperature.

“These attacks coming from Westminster against Europe and the European Commission will not be responded to in the same way by Europe and the European Commission, although I would like to do it.”

Mr Tusk added: “We have treated Prime Minister May with the greatest respect, all of us.

“And we really appreciate the efforts by the Prime Minister to ratify our common agreement.

“My impression is that in fact we have treated Prime Minister May with much greater empathy and respect than some British MPs, for sure.”

Luxembourg prime minister Xavier Bettel said Mrs May had won the “best possible deal” and that it was now up to MPs at Westminster to be “responsible” and vote for it.

“Now the MPs in London should be responsible and know if they want to have the best possible deal or to go in the direction where they don’t know what will come out” he said.

“They should not forget they are elected to represent their citizens and not to just have their personal, political future thinking about that.”

DUP leader Arlene Foster said the reaction of the EU to Mrs May’s appeals for help was “unsurprising”.

“They are doing what they always do.

“The key question is whether the Prime Minister will stand up to them or whether she will roll over as has happened previously,” she said.

Labour leader Jeremy Corbyn said: “The last 24 hours have confirmed that Theresa May’s Brexit deal is dead in the water.

“The Prime Minister has utterly failed in her attempts to deliver any meaningful changes to her botched deal.”