European Union leaders have said the Brexit withdrawal agreement is "not open for renegotiation", despite appeals from Theresa May.
She wanted legal assurances on the Irish backstop to help her deal get through Parliament, after she delayed a Commons vote in anticipation of defeat.
The PM said the deal was "at risk" if MPs' concerns could not be addressed.
European Commission president Jean-Claude Juncker said there could be clarifications but no renegotiation.
He urged the UK to set out more clearly what it wants, adding that the commission will publish information on 19 December on its preparations for a no-deal Brexit.
Mrs May returned to Brussels for a summit with EU leaders on Thursday, at the end of a week that has seen her first delay the vote on the withdrawal agreement in Parliament, then win a vote of confidence brought by MPs unhappy with it.
She vowed to listen to the concerns of the 37% of Tory MPs who voted against her and was hoping to "assuage" their concerns about the controversial "backstop" plan in the agreement.
Critics say the backstop - aimed at preventing a hard border in Northern Ireland - would keep the UK tied to EU rules indefinitely and curb its ability to strike trade deals.
Conservative MPs 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.
At the summit, Mrs May was seeking legal assurances that the backstop, if used, would be temporary.
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."
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 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 the Brussels summit, Mr Tusk called the backstop "an insurance policy," saying it was the EU's "firm determination" to work "speedily" on alternative arrangements.
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".
Mr Juncker urged the UK to tell the EU what it wants in the future relationship.
"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 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.
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".