Brussels (CNN) The European Union will not throw British Prime Minister Theresa May a lifeline over Brexit, after she survived a bruising confidence motion triggered by members of her own party.

Leaving Westminster convulsed by the crisis over her leadership, May came to Brussels to plead with EU leaders Thursday at a summit to make the agreement more palatable to skeptical lawmakers in London.

As she arrived, May said she was there to speak with EU leaders about what it will take to "get this deal over the line."

But following the meeting, representatives of the bloc said they were sticking with the withdrawal agreement previously agreed with London, including the Northern Ireland backstop deal which threatens to derail it in the UK Parliament. May pulled a vote on the deal this week when it became clear she would suffer a heavy defeat.

"If we go into negotiations on the future relationship, we need to have a well-constructed proposal and a cogent idea from our British partners and friends, and then we'll look at that," European Commission President Jean-Claude Juncker told reporters.

May had been seeking legally enforceable guarantees surrounding the Irish backstop -- the insurance policy designed to prevent the return of a hard border between Northern Ireland and the Republic of Ireland. The backstop has emerged as the crucial sticking point for many in May's Conservative Party, furious that Britain could only leave it with the approval of the EU.

Hardline Brexiters have argued that the risk of a "no deal" exit -- which would be devastating for the UK economy and British infrastructure, but would also damage the EU -- would compel Brussels to compromise, but EU leaders don't appear ready to blink.

"In terms of a no deal, let's not forget ultimately, it is within the gift of the UK government and the UK parliament to take the threat of no deal off the table," Irish Prime Minister Leo Varadkar said Thursday.

"It is possible if the UK wishes to, to revoke Article 50, or, if that's a step too far to seek an extension of Article 50," he added, referring to the piece of EU law by which the UK triggered Brexit.

UK Prime Minister Theresa May arrives at the EU leaders' summit in Brussels on Thursday.

May to 27: 'We must get this right'

"We must get this right," May told the EU27 on Thursday. "Let's hold nothing in reserve. Let's work together intensively to get this deal over the line in best interest of all our people."

She said a route can be found for the Brexit deal to be approved by the House of Commons.

"We have to change the perception that the backstop could be a trap from which the UK could not escape," she said. "Until we do, the deal, our deal, is at risk... the best result for all of us is to get this deal delivered in an orderly way and get it done now."

May called on leaders to trust her track record.

"Over the last two years, I hope I have shown you can trust me to do what is right, not always what is easy, however difficult that might be for me politically. ... With the right assurances this deal can be passed. Indeed it is the only deal that is capable of getting through my parliament."

Earlier Thursday, a Downing Street spokesperson confirmed to CNN that British MPs will not have to vote on May's Brexit deal before the end of the year. However, Downing Street has assured that it will take place before January 21, 2019.

Speaking after a meeting with May, Irish PM Varadkar told reporters that the "deal that we have is the only deal on the table" and made clear that he considered it the United Kingdom's responsibility to resolve the crisis.

"I think there is one thing that's undeniable. All these difficulties that Europe now faces, not least Ireland, but all Europe, we now face these difficulties because of a decision the UK made to leave the European Union," he said. "We respect the decision they've made, but it does mean there is an obligation now on them to come up with the solutions."

While May won Wednesday's confidence vote, by 200 votes to 117, the margin of victory was significantly narrower than her supporters expected and she arrived in Brussels with her authority further dented.

An EU diplomat told CNN that the other 27 EU nations were looking to May to "convince" them she can get the withdrawal agreement through the UK Parliament.

"We've seen what happened yesterday. Convince us that what you ask will make a difference. If she pulls that off then we can talk... in the end they are politicians and they will want to help her. We are ready to be convinced," the diplomat said.

The 27 are "expecting serious clarification of what she plans to do," the source said, but for now, May appears to be "playing for time" in the hopes that uncertainty will win over the lawmakers she needs to back her deal.

The diplomat added that they are very concerned about an accidental hard Brexit, saying the "most likely scenario is stumbling into a no deal."

Why is the Irish border backstop such a big issue? One of the big fears in the Brexit debate is that Britain's departure from the EU will mean the reintroduction of border posts on the frontier between Northern Ireland, which remains part of the UK, and the Republic of Ireland, an EU member. Border infrastructure was often targeted by Irish nationalist paramilitaries during the "Troubles" -- the 40-year sectarian conflict in which more than 3,500 people died. UK Prime Minister Theresa May's Brexit deal avoids the reintroduction of a so-called "hard" border, because of the built-in transition period that keeps the UK in a customs union with the EU. The problems come if there's no agreement on what to do after the transition period ends in 2020. Enter the Irish "backstop," an insurance policy designed to avoid a hard border if no other solutions are found by that time. This has infuriated hard-line Brexiteers, worried the UK will never "properly" leave the bloc. They want to be free of the customs union, in order to forge international trade deals that would require the UK to be free of EU regulations on issues like agriculture, fisheries, food standards and the environment. The Democratic Unionist Party, whose 10 MPs prop up May's minority government in London, also demands a clean break with the EU, and oppose any move to give Northern Ireland different status from mainland Britain. The big sticking point is that the Brexit deal, as it stands, states that neither side can leave the backstop unilaterally. Brexiteers hate the idea that the EU would hold a power of veto over the UK. But others, including the Irish government, argue that the backstop would be meaningless if Britain could tear it up at will.

Europeans are holding firm

Arriving in Brussels, EU leaders suggested they could offer greater clarity around the deal -- but there was little sign they would make changes substantial enough to win over May's critics at home.

"The thing now tonight is we have to seek clarifications, particularly on the backstop," said Dutch Prime Minister Mark Rutte. "There is this whole thinking in the UK that this backstop is inevitable, that it will be triggered. I can assure you one thing, there is nobody in their right mind in the European Union who wants to trigger the backstop because this is bad news not only for the UK but also for the EU."

Rutte added that it would be "impossible to break open" the withdrawal agreement, saying: "This is the only deal possible on the table."

Austrian Chancellor Sebastian Kurz echoed that message. "The deal we already have is a good one. I think there is also an understanding from Theresa May that there will be no new negotiation of the withdrawal agreement," he said.

"But, of course ... I think there will be some readiness from our side to maybe find some better explanation about the future relationship ... There is also some room to have a better interpretation of what we agreed on."

French President Emmanuel Macron also rejected any renegotiation of the deal. "There can be a political discussion, but not a legal one," he said. "It is up to Theresa May to say what the political solution is to get a majority on this agreement."

However, José Manuel Barroso, former president of the European Commission, took to Twitter to urge the European Council to show some flexibility over the backstop issue, saying that there would be no more important future relationship for either the UK or EU than with each other.

I know it's possible some flexibility of the #EU to get a deal acceptable by the #uk . If there's political will the excellent lawyers of the #EU can find a creative solution for the #backstop issue. And wisdom should prevail btw the EU and U.K. — José Manuel Barroso (@JMDBarroso) December 13, 2018

Backstop challenge

The size of Wednesday's rebellion underscored the daunting task faced by the Prime Minister if she is to secure approval in a divided House of Commons for her Brexit deal.

Hardcore Brexiters' opposition to the agreement has crystallized around the Northern Ireland backstop.

But Brexit Secretary Stephen Barclay insisted in an interview with the BBC's Radio 4 on Thursday that any alternative deal to the one negotiated by May would also require a backstop on Northern Ireland.

May acknowledged that Wednesday had been "a difficult day" and said she was "grateful" for the support she'd received. "But I've also heard loud and clear the concerns of those who didn't feel able to support me," she said in Brussels.

May confirmed that she did not intend to lead her party indefinitely. "I said in my heart I would love to be able to lead the Conservative Party into the next general election, but I think it is right that the party feels it would prefer to go into that election with a new leader."

Conservative Brexiter Jacob Rees-Mogg, who led the rebellion against May, has urged her to resign, saying the country needs a new leader.