Theresa May has told EU leaders the withdrawal deal they have negotiated with the UK is “at risk”, if MPs’ concerns cannot be addressed.
The PM, who delayed a Commons vote on the deal in anticipation of a defeat, said the agreement “can be passed”.
She urged EU leaders to work with her to “change the perception” of the controversial “backstop” plan.
But European Council president Donald Tusk said the withdrawal agreement was “not open for renegotiation”.
Speaking after a summit in Brussels, Mr Tusk called the backstop – aimed at preventing a hard border in Northern Ireland – “an insurance policy,” saying it was the EU’s “firm determination” to work “speedily” on alternative arrangements.
Mrs May had travelled to the summit of EU leaders to seek legal assurances from them that the backstop, if used, would be temporary.
Her return to Brussels followed a confidence vote on her leadership on Wednesday – brought by MPs unhappy with her Brexit policy – which she won.
She vowed to listen to the concerns of the 37% of Tory MPs who voted against her and is hoping to “assuage” their concerns about the “backstop” plan in the withdrawal agreement.
Critics say it would keep the UK tied to EU rules indefinitely and curb its ability to strike trade deals.
Conservative MPs have demanded changes to the backstop to make it clear that it could not last forever, and the UK could terminate the arrangement on its own.
The BBC’s political editor Laura Kuenssberg said one of the ideas being considered in Brussels was whether they could agree a “start date” for a future trade relationship between the UK and the EU after Brexit instead of an end date for the backstop.
In comments released by Downing Street on Thursday, Mrs May told EU leaders she firmly believed the deal could get through the Commons and told them: “Let’s work together intensively to get this deal over the line in the best interests of all our people.”
But she added: “We have to change the perception that the backstop could be a trap from which the UK could not escape. Until we do, the deal – our deal – is at risk.”
And she said it was in everyone’s interests for the deal to be “delivered in an orderly way and to get it done now” rather than “to run the risk of an accidental ‘no deal’ with all the disruption that would bring or to allow this to drag on any further”.
“There is a majority in my Parliament who want to leave with a deal so with the right assurances this deal can be passed. Indeed it is the only deal capable of getting through my Parliament,” she said.
“Over the last two years, I hope that I have shown you that you can trust me to do what is right, not always what is easy, however that difficult that might be for me politically.”
Mrs May took questions from the leaders of the 27 other EU states before leaving while they discussed their response over dinner.
Following the meeting, Mr Tusk said the backstop would “apply temporarily unless and until it is superseded by a subsequent agreement that ensures that the hard border is avoided”.
The European Council’s conclusions on Brexit – published on Thursday evening – say the EU would use its “best endeavours to negotiate and conclude expeditiously a subsequent agreement that would replace the backstop, and would expect the same of the United Kingdom, so that the backstop would only be in place for as long as strictly necessary.”
In other words, the EU would continue trying to negotiate a trade deal with the UK even if the Irish backstop had been triggered at the end of the transition period.
The Brexit withdrawal agreement only talks about “best endeavours” being used to reach an agreement during the transition period.
European Commission president Jean-Claude Juncker said clarifications could be added, “but there will be no renegotiations”.
He urged the UK to tell the EU what it wants in the future relationship, saying: “Our UK friends need to say what they want, instead of asking us to say what we want and so we would like within a few weeks our UK friends to set out their expectations for us, because this debate is sometimes nebulous and imprecise and I would like clarifications”.
He said the commission will publish information on its preparations for a no-deal Brexit on 19 December.
Austrian Chancellor Sebastian Kurz, who holds the rotating EU presidency, suggested there could be a special Brexit summit in January to agree “additional assurances”.
But Irish premier Leo Varadkar said that while EU was keen to be “helpful”, some of the suggestions she had put forward were “difficult” and warned there could be no “unilateral exit clause” on the backstop.
Downing Street has confirmed that MPs will not now vote on Mrs May’s deal before Christmas and said the vote would happen “as soon as possible in January”.