The prime minister is expected to plead with EU leaders to drop their Irish backstop proposal at a make-or-break summit dinner on Wednesday night after seeking the support of members of her cabinet on Tuesday morning.

With time running out before Wednesday’s meeting, May used an emergency Commons statement to say the EU’s plan “threatens the integrity of our United Kingdom” because it could lead to the creation of a customs border in the Irish Sea.

She told an audience of largely sceptical MPs that the EU had stuck to its backstop proposal because Michel Barnier’s negotiating team had told her there was not time to evaluate a British UK-wide counter-proposal “in the next few weeks”.

The prime minister was due to speak to the French president, Emmanuel Macron, on Monday night as she tries to lobby EU leaders to change their minds. May has already also spoken to the German chancellor, Angela Merkel, and the Dutch prime minister, Mark Rutte, in recent days, according to No 10.

A group of eight Brexiter ministers met on Monday night at a meeting dubbed the “pizza summit”, organised by Andrea Leadsom, the leader of the house, to discuss May’s strategy before Tuesday’s cabinet meeting, amid separate concerns from the Tory right that May’s all-UK backstop plan needed to be clearly time limited.

The strength of the turnout – including the Brexit secretary, Dominic Raab, the foreign secretary, Jeremy Hunt, and the environment secretary, Michael Gove – is likely to concern Downing Street. They are understood to have aired concerns about May’s negotiating strategy, although one of those present said no strategy for the cabinet meeting was agreed.

Friends of another one of those attending, Penny Mordaunt, the international development secretary, said she was remaining loyal to the prime minister for now. Others present included the transport secretary, Chris Grayling, the attorney general, Geoffrey Cox, and the chief secretary to the Treasury, Liz Truss.

EU leaders acknowledged that the Brexit talks had hit a roadblock, although some insisted the problems could still be overcome. Donald Tusk, the EU council president, said a no-deal scenario was “more likely than ever before”.

Merkel said: “We were actually pretty hopeful that we would manage to seal an exit agreement … at the moment, it looks a bit more difficult again”. Speaking to the German Foreign Trade Federation, she said a breakthrough was still possible but would need “quite a bit of finesse and if we aren’t successful this week, we’ll just have to keep negotiating”.

The cautiously optimistic tone was further echoed by Macron, who had demanded “maximum progress” by the time of this week’s leaders’ summit to allow an extraordinary Brexit summit to be called in mid-November. “I believe in our collective intelligence, so I think we can make progress,” he said.

A backstop is required to ensure that there is no hard border in Ireland if a comprehensive free trade deal cannot be signed before the end of 2020. The EU plan would mean that Northern Ireland would remain in the single market and the customs union, prompting fierce objections from Tory hard Brexiters and the Democratic Unionist party, which props up her government.

That prompted May to propose a country-wide alternative in which the whole UK would remain in the parts of the customs union after Brexit, but she admitted in the Commons that, despite months of talks, her counter-proposal had not been accepted.

“The EU still requires a ‘backstop to the backstop’ – effectively an insurance policy for the insurance policy. And they want this to be the Northern Ireland-only solution that they had previously proposed,” May told MPs on Monday.

Raising the stakes, the prime minister insisted that the EU’s insistence amounted to a threat to the constitution of the UK: “We have been clear that we cannot agree to anything that threatens the integrity of our United Kingdom,” she added.

The prime minister insisted that she was demanding of the EU that the UK backstop was time limited: “I need to be able to look the British people in the eye and say this backstop is a temporary solution.”

May had gone to the Commons to clarify the status of the Brexit negotiations a day after a deal had been thought to be close. The Brexit secretary, Dominic Raab, was dispatched to Brussels on Sunday afternoon, only to return empty-handed with No 10 warning that the talks had reached an impasse.

Few MPs in the Commons spoke in support of May. Tory Brexiters, led by former cabinet ministers Iain Duncan Smith, repeatedly pressed her to confirm there would be a specific end date for the temporary backstop plans.

The prime minister avoided answering the question, telling MPs: “I continue to believe that we should be working to ensure that the backstop never does come into place.”

Simon Clarke, the Conservative Brexiter MP for Middlesbrough South, told her she had “failed to reassure the house”.

The prime minister also came under pressure from the remain wing of her party. The former home secretary Amber Rudd urged her to deliver a Brexit that also worked for the 48% who voted to remain and the former education secretary Nicky Morgan warned may there was no majority for no deal in the Commons and that MPs would have to “step in” if she failed to get one.

May highlighted a concession she had already made on the EU withdrawal bill, telling MPs: “If it were the case that at the end of the negotiation process actually it as a no deal … then that would come back to this house and then we would see what position this house would take in the circumstances”.

About 15 MPs, including four Conservatives, used the debate to urge May to reconsider holding a second referendum. Former cabinet minister Dominic Grieve, who has led previous rebellions against the prime minister’s plans, said he would not back the transition period, which he described as a “condition of vassalage”, unless there was another vote.

Nigel Dodds, the DUP’s Westminster leader, pressed May to reiterate the UK would leave the EU “together with no part hived off either in the single market or customs union differences”.

Dodds was visibly unhappy with May’s answer, shaking his head when she replied in general terms: “We will be leaving the European Union together.”

Jeremy Corbyn urged May to “put the country before her party” and stand up to the “reckless voices” on the Tory benches. “It is clear that the prime minister’s failure to stand up to the warring factions of her own side have led to this impasse.”

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