Theresa May has revealed that Donald Trump advised her to “sue the European Union” rather than negotiate with the 27-country bloc, in a private conversation that the US president referred to during his visit to the UK on Friday.

The prime minister was asked on the BBC’s The Andrew Marr Show what the “brutal” Brexit negotiating advice was that Trump had talked about in their joint press conference outside the prime minister’s Chequers country retreat.

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Revealing it for the first time, May said: “He told me I should sue the EU.” After being prompted by a surprised Marr, May repeated: “Sue the EU, not go into negotiations with them, sue them.”

The prime minister smiled, and indicated she had disregarded the advice, saying “actually we’re going into negotiations with them”, in remarks that will be interpreted as a put-down of the president. Trump leaves the UK this afternoon to fly to Helsinki to meet the Russian president, Vladimir Putin.

On Friday, Trump had said he gave May “a suggestion, I wouldn’t say advice” about how to handle the Brexit talks, without revealing what it was. “I think she found it maybe too brutal, and that’s OK. I gave her a suggestion, not advice. I could fully understand why she thought it was a little bit tough.”

Marr’s interview with May comes at a politically perilous moment for the prime minister, who is facing calls for a confidence vote over her leadership of the Conservative party, and the prospect of Boris Johnson spelling out the reasons for his resignation in a newspaper column on Monday.

Reports at the weekend suggested that about 40 of the 48 MPs needed had lodged no-confidence letters with the chairman of the party’s backbench 1922 Committee after May had indicated that the UK was prepared to sign up to a “common rulebook” for food and goods after Brexit.

May indicated she would seek to fight off any challenge, saying: “I want to focus people’s minds on how you ensure you achieve that prize, the benefits of leaving the European Union.” She added: “I have always said I’m in this for the long term.”

Jacob Rees-Mogg, who chairs the hard-Brexit European Research Group (ERG), voiced concern over May’s policy in a BBC interview shortly after the prime minister’s appearance. “This seems to be a hopeless way to negotiate, to accept what the other side says at an early stage of negotiations as holy writ.” He added: “This is why I think she is a remainer, who has remained a remainer.”

But Rees-Mogg repeatedly declined to call for a change of leader, instead saying it was necessary for May to change policy. He said ERG members would mount a show of strength on Monday evening, by voting for hard-Brexit amendments on the customs bill. “The inevitable consequence of the parliamentary arithmetic,” he said, “is she will need to change it [Brexit policy] if she is to keep the party united.”

The prime minister was initially embarrassed by Trump when he gave an interview to the Sun, published on the morning of the press conference, in which he appeared to say that May’s Brexit plan would prevent a trade deal with the US and said Johnson would make a good prime minister.

Trump subsequently apologised to May in private and partially backtracked on the remarks on Friday, saying that the UK should pursue its own Brexit policy and trade negotiations but: “Just make sure you can trade with us.”

Marr also asked May if she believed Trump had “a medical problem” with stairs because he frequently takes her hand when the pair find themselves facing a step or two. May acknowledged that “whenever he takes me down a slope, or stairs ... he takes my hand to help in going up the steps”.

The BBC interviewer then asked her if that made her look submissive. “Oh, Andrew come on,” May replied. “Are you telling me you have never said to somebody: let me help you?” 

She also said she believed she could trust Trump at his word, and in an echo of language used by the president, added that the UK continued to enjoy “the highest degree of special relationship with the United States”.