British Prime Minister Theresa May has appealed to EU leaders to give her the assurances she needs to get the backing of MPs for her Brexit deal.

At the EU summit in Brussels, she insisted she could construct a Commons majority for the controversial Withdrawal Agreement, despite heavy criticism from all sides.

However, in a meeting with the leaders of the remaining 27, she said she had to be able to convince MPs the UK would not find itself tied to the EU indefinitely through the Northern Ireland "backstop".

"There is a majority in my parliament who want to leave with a deal so with the right assurances this deal can be passed," she said in prepared remarks released by her office.

"Indeed it is the only deal capable of getting through my parliament."

The prime minister, who yesterday survived a bruising vote of no confidence by Tory MPs, said a package of assurances around the backstop could "change the dynamic" at Westminster.

At the same time, she made clear a failure by EU leaders to offer concessions risked the collapse of the whole agreement with the UK leaving in March in a disorderly, no-deal Brexit.

"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," she said.

"I am in no doubt that the best result for all of us is to get this deal delivered in an orderly way and to get it done now.

"It is in none of our interests to run the risk of accidental no-deal with all the disruption that would bring, or to allow this to drag on any further."

She ended with a highly personal appeal to EU leaders to put their trust in her to deliver on her promises and to give her the political room for manoeuvre she needs.

"Over the last few 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," she said.

"We must get this right and hold nothing in reserve. Let's work together intensively to get this deal over the line in the best interests of all our people."

Negotiators said the leaders would listen to Mrs May and see what they can offer her in terms of a statement, but insist the 585-page Withdrawal Agreement itself cannot be renegotiated.

Austrian Chancellor Sebastian Kurz said: "The discussions in Britain ... do not make the situation any easier, but we should nevertheless do everything we can to avoid a hard Brexit." 

Austria holds the rotating EU presidency and Mr Kurz arrived in Brussels ahead of the summit to talk with host Donald Tusk, the president of the European Council, about the agenda.

"It is clear that the Withdrawal Agreement will not be opened up and amended but there are, of course, other ways of moving towards one another that are in the interests of both sides," Mr Kurz said.

But determined to remain loyal to continuing member Ireland, European officials insist in public and in private that the backstop must stay. "The idea of a sell-by date won't stand," one said.

And any modification or addendum to an agreement on plans for future UK-EU relations will not be legally binding, negotiators say.

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Anything Mrs May does come away with must convince her own party, its DUP allies and a majority of British MPs to back the deal if and when it returns to Westminster.

If the agreement is still found wanting not only would Britain and its main trading partners face economic chaos, but Mrs May could be finished off by a parliamentary vote of no confidence.

Mrs May won the backing of 200 Conservative MPs, but 117 voted to oust her in a secret no confidence ballot last night - and only after she had conceded she would step down before 2022 elections.

She said she wanted to "get on with the job of delivering Brexit", and to see "politicians on all sides coming together".

But her hardline opponents claimed the vote against her had been higher than expected and that most backbench members, who do not hold jobs in Mrs May's government, want her gone.

"She ought to go and see the Queen and resign urgently," leading Brexit rebel Jacob Rees-Mogg told the BBC.

Eurosceptics hate the divorce deal agreed last month, which they fear risks tying Britain to the EU for years after Brexit.

Their confidence vote followed the prime minister's decision on Monday to postpone a planned vote in the House of Commons in the face of what she feared would be a crushing defeat.

She has promised to hold that vote before 21 January - when she may yet still lose, plunging the Brexit process into fresh crisis.

Additional reporting PA