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Theresa May will update Parliament later on her latest round of Brexit talks, as she continues to try to gather backing for her deal.
The PM is due to make a statement to MPs, ahead of a debate on Thursday on the next steps for Brexit.
She visited Dublin and Brussels last week seeking changes to the backstop – the “insurance” policy to avoid a hard border on the island of Ireland.
The EU has reiterated it will not renegotiate the withdrawal agreement.
Speaking after a dinner with Brexit Secretary Steve Barclay on Monday night, EU negotiator Michel Barnier said the talks had been “constructive”, but added it was “clear from our side we’re not going to reopen the withdrawal agreement, but we will continue our discussions in the coming days”.
Mr Barclay and Mr Barnier had agreed to further talks “in the coming days”, a statement from the Department for Exiting the European Union said.
Their teams would continue to work in the meantime “on finding a way forward”, it added.
Mrs May’s statement to the House of Commons is being made a day earlier than had been expected.
It comes after MPs voted last month for the PM to find alternatives to the current backstop.
By Laura Kuenssberg, BBC political editor
Look away now if you are of a nervous disposition.
This week’s Brexit votes may not come to much.
The prime minister is essentially asking for more time to get something to show for her promise of changes to the controversial backstop.
That might mean that she’s back before the end of this month with something concrete to vote on.
Or it may well not.
I know that sounds nebulous, a word you might choose to use.
But there’s a building impression in Westminster, and in Brussels, that there simply won’t be a resolution to these months of indecision until weeks before, maybe even days before we are due to leave (that’s why some cabinet ministers believe that, even though it’s not desirable, the government may end up asking to extend the whole process as we discussed here before).
It also follows an exchange of letters between Mrs May and Labour leader Jeremy Corbyn.
The Labour leader wrote to the PM on Wednesday with a list of five demands to secure his party’s support for her deal, including a permanent customs union.
The prime minister struck a conciliatory tone in her response overnight and said she looked forward to the two parties meeting again “as soon as possible” to discuss ways forward on Brexit.
But No 10 said it rejected any proposals to remain in a customs union with the EU.
Some of her cabinet members also quashed the idea, with International Trade Secretary Liam Fox calling Mr Corbyn’s proposal a “dangerous delusion”.
Mrs May told the Labour leader: “It is good to see that we agree that the UK should leave the European Union with a deal and that the urgent task at hand is to find a deal that honours our commitments to the people of Northern Ireland, can command support in Parliament and can be negotiated with the EU – not to seek an election or second referendum.”
This is despite Mr Corbyn repeatedly saying there should be a general election if Mrs May cannot get a deal through Parliament. He has also faced pressure from some of his MPs to push for another public vote on Brexit.
Shadow Brexit secretary Keir Starmer said his “key question” about Mrs May’s response was: “Is she prepared to move her red lines and find a consensus?”
“I don’t see that in the letter,” he said.
“The point of the exercise was to say, look, there is a majority for a close economic relationship, if you’re prepared to try to find it, and I’ve said for some time we should test that by having a vote on a customs union.”
Labour MP Lisa Nandy told the BBC’s Politics Live that there were between 40 and 60 of her colleagues “who are actively looking for ways to support” a revised Brexit deal.
She said the government needed to “get serious” about policy on a customs union and guarantee to “legislate for the protections around workers’ rights”.
Mr Corbyn’s letter to the prime minister was “interesting in tone and substance”, Mr Barnier said.
Mr Barclay met a group of Tory MPs on Monday, ahead of his trip to Brussels, who are working on possible alternatives to the backstop.
Members of the Alternative Arrangements Working Group – including Conservative MPs Steve Baker, Marcus Fysh, Owen Paterson, Damian Green and Nicky Morgan – attended the meeting.
Mr Baker said the talks had been “constructive”.