By Gabriela Baczynska and Elizabeth Piper
BRUSSELS (Reuters) - Britain's weakened prime minister, Theresa May, appealed to fellow EU leaders on Thursday for concessions to help her win support in parliament next month for a deal that can smooth Britain's exit from the European Union.
"Trust me," she urged the other 27 leaders, according to a British official. She assured them she could get the deal they agreed with her last month through parliament -- if they would only help her overcome British resistance to its terms.
"It is in none of our interests 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," she warned, before leaving the others to discuss their response over dinner.
Twenty-four hours after surviving an internal party vote that threatened to oust her, May was met largely by readiness to help from European Union leaders but also by a warning: we will not reopen the divorce agreement struck last month.
Leaders who have said that some kind of declaratory text aimed at easing British concerns could be delivered on Thursday, with possibly more to come next month before May puts her package to lawmakers.
Diplomats said one option would be to put a date -- possibly end-2021 -- as a non-binding target for concluding a close EU-UK trade alliance that would end the need for the "Irish backstop" -- a mechanism anathema to May's party.
However, even introducing an aspirational date would be hard for some, including Dublin, to accept, as May struggles to find a way to satisfy her critics' demand for a strict time-limit.
The backstop could leave Britain tied forever to EU customs rules until a better way is agreed to ensure there are no disruptive frontier controls on Northern Ireland's land border with the Irish republic.
"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," May told the other leaders.
"There is a majority in my parliament who want to leave with a deal so with the right assurances this deal can be passed."
DEADLOCK
Britain's departure from the EU, its biggest shift in trade and foreign policy for over 40 years, is proving far from easy, complicated by the deep divisions in her Conservative Party.
With less than four months before Britain is due to leave on March 29, May faces deadlock in parliament over the deal, which has hardened positions at home, throwing up more uncertainty for businesses trying to predict what will happen to the economy.
While others tried to temper their language by expressing a desire to help May, French President Emmanuel Macron was blunt.
"We cannot reopen a legal agreement, we can't renegotiate what was negotiated for several months," he told reporters. "It's up to Theresa May to tell us what political solution she expects to pursue to find a majority for this deal."
German Chancellor Angela Merkel was less strident, saying: "We can of course talk about whether there are additional assurances but in this the 27 EU members are together and will make their interests clear, although always in the spirit that we want very, very good relations with Great Britain after Great Britain has left the European Union."
MAY SURVIVES
May won Wednesday's party ballot 200-117 but the size of the vote against her deepened divisions just weeks before parliament needs to approve a deal to prevent a disorderly exit from the EU. Victory also came at a price - May promised she would step down by the next election scheduled for 2022.
Her spokeswoman said that the Brexit deal would be put before parliament "as soon as possible" in January.
The level of opposition to her deal was underlined earlier this week when May was forced to delay a parliamentary vote on her deal for fear of a defeat.
May wants legal assurances that the backstop will not remain in place indefinitely.
Diplomats expect it to change after May tells the other national leaders what she needs. They suggested the EU may be readying more solid assurances for May in January.
"Today is about de-mystifying this whole Northern Ireland backstop. Nobody in the EU wants to use it. But we need to have it," said Dutch Prime Minister Mark Rutte.