THERESA May and Jean-Claude Juncker became involved in a public argument on the last day of a crucial EU summit in Brussels as she sought concessions on her Brexit deal.

The Prime Minister was caught on camera yesterday morning angrily confronting the European Commission president after he accused her of not knowing what she wants for Brexit.

Juncker, who had addressed EU leaders at the summit on Thursday evening, requested clarity from the UK. He said: “Our UK friends need to say what they want, instead of asking us to say what we want ... because this debate is sometimes nebulous and imprecise.”

But his remarks annoyed May, who took issue with him. Official video footage showed a tense exchange. According to lip readers, May said: “You called me nebulous. Yes you did!”

Juncker, who had one hand on May’s arm, looked to be trying to soothe her anger by denying making the jibe. A man, believed to be Dutch premier Mark Rutte, is then seen walking toward the pair in a bid to break up the discussion.

At a later press conference May and Juncker were keen to play down the exchange. Asked about what she had said to him, May told reporters: “I had a robust discussion with Jean-Claude Juncker – I think that’s the sort of discussion you’re able to have when you have developed a working relationship and you work well together.

“And what came out of that was his clarity that actually he’d been talking – when he used that particular phrase – he’d been talking about a general level of debate.”

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Juncker denied describing May as “nebulous” and said the word referred to the Brexit debate coming out of the UK. He stressed EU leaders had “the highest respect” for May and sympathised with her over the threat to her position from her own MPs. He said May was “kissing me” after she “checked” the comments he made on Thursday night.

“She thought that I did criticise her by saying yesterday night that the British position was nebulous ... I did not refer to her but to the overall state of the debate in Britain,” he said.

He added: “I was not addressing her, and in the course of the morning after having checked what I said yesterday night, she was kissing me.”

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Peacemaker: Donald Tusk moved to reassure the Prime Minister

European Council president Donald Tusk also stepped in. “My impression is that in fact we have treated Prime Minister May with much greater empathy and respect than some British MPs, for sure,” he said in reference to the confidence vote called by Tory MPs in May.

The bizarre turn of events overshadowed the second day of the summit, which ended with the announcement the two sides would have further talks in the coming days about how May can obtain “further assurances” which might persuade MPs to back the Brexit withdrawal agreement.

In a press conference at the end the summit May acknowledged that it would not be possible to reopen the agreement to alter the backstop provisions which sparked the rebellion among Tory MPs.

May welcomed the official conclusions issued by the EU27 which committed the EU to trying to get a post-Brexit trade deal agreed quickly enough to avoid the need for a backstop to keep the Irish border open. “As formal conclusions, these commitments have legal status and therefore should be welcomed,” she said. But she added: “The EU is clear, as I am, that if we are going to leave with a deal, this is it.

“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.”

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German chancellor Angela Merkel insisted it was for Britain to take the next steps, saying: “The 27 member states have given assurances. They are contained in the conclusions of yesterday evening.”That is our position. That is what we have put on the table. And now we expect Great Britain to respond.”

Michael Gove said the Prime Minister’s Brexit deal will survive. Speaking after the Journalist’s Charity Scotland lunch in Glasgow, Gove was asked if the deal can pass. He said: “Yes I think it can. The Prime Minister has been conducting negotiations on behalf of the country with the tenacity, determination and focus on the national interest that has always characterised her approach.

He added: “And absolutely we will have an opportunity to hear from the Prime Minister when she is speaking in the House of Commons next week and I know that she will have the enthusiastic support support of the Cabinet for the course that she is on.”