Ministers are facing MPs’ questions about the failure to strike a Brexit deal following a DUP backlash.
The DUP, whose support the government needs in key votes, said it would not accept a deal on the Irish border which saw Northern Ireland treated differently from the rest of the UK.
The UK is seeking permission from Brussels to open trade talks.
Brexit Secretary David Davis is responding to Labour’s urgent question in the Commons.
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The UK is due to leave the EU in March 2019 having voted for Brexit in last year’s referendum.
Mrs May wants to begin drawing up a deal on post-Brexit trade relations between the UK and the EU but first needs to show “sufficient progress” has been made so far on “divorce” issues before European leaders meet on 14 December.
The three issues that need to be resolved are the Northern Ireland border, citizens’ rights and the amount of money the UK will pay as it leaves.
Talks between Prime Minister Theresa May and European Commission president Jean-Claude Juncker broke up without agreement on Monday, after the Democratic Unionist Party (DUP) objected to a draft agreement on the future of the Irish border.
Key to the row is how closely aligned Northern Ireland’s regulations will be with those of the Republic of Ireland, and the rest of the EU, in order to avoid a “hard” border.
The prime minister needs the support of the DUP, which is Northern Ireland’s largest party and has 10 MPs at Westminster, because she does not have a majority to win votes in the House of Commons.
DUP leader Arlene Foster was invited to hold talks with Mrs May in London on Tuesday, but the party’s Westminster leader will meet the government’s chief whip instead.
The EU is treating the row as a “domestic British political issue”, BBC Brussels correspondent Adam Fleming said.
“The show is now in London,” said a European Commission spokesman.
Reacting to leaks of the draft agreement, Scottish First Minister Nicola Sturgeon said that if Northern Ireland was allowed to operate under different rules there was “surely no good practical reason” why other parts of the UK could not do the same – a message echoed by Welsh First Minister Carwyn Jones and London Mayor Sadiq Khan.
Ms Sturgeon appealed to other parties to keep the UK in the EU’s customs union.
This could be the moment for opposition and soft Brexit/remain Tories to force a different, less damaging approach – keep the UK in the single market and customs union. But it needs Labour to get its act together. How about it @jeremycorbyn?
— Nicola Sturgeon (@NicolaSturgeon) December 5, 2017
End of Twitter post by @NicolaSturgeon
Scottish Conservative leader Ruth Davidson called for a UK-wide deal, rather than specific arrangements for Northern Ireland.
On the BBC’s Today programme, former Brexit minister David Jones described Monday as a “very difficult day” and said the prime minister now had “a lot of talking to do” with DUP leader Arlene Foster.
He believed the DUP had been spoken to beforehand about what was being proposed, but that the precise wording had not been explained and the Northern Ireland party “clearly thought twice”.
Mr Jones, one of the Eurosceptic MPs on the Conservative benches, said he too was not happy with the key phrase the DUP objected to in the draft agreement, which apparently guaranteed “regulatory alignment” between Northern Ireland and the Republic of Ireland.
Former top Foreign Office official Lord Ricketts said the row was “damaging” for Mrs May, adding that it was “pretty extraordinary that this wasn’t all stitched up with the DUP beforehand”.
The Republic of Ireland’s Prime Minister Leo Varadkar is understood to have insisted on the clause to prevent the return of a “hard border” on the island of Ireland, amid concern it could undermine the 1998 peace treaty – the Good Friday Agreement that brought an end to 30 years of sectarian conflict in Northern Ireland.
But the DUP says it would prevent Northern Ireland from leaving the EU “on the same terms” as the rest of the UK.
Leo Varadkar said he had been “surprised and disappointed” by the failure to get a deal on Monday and claimed the UK had changed its mind at the last minute after the DUP raised objections.
But it said the border was not the only outstanding problem and disagreement remains over the role of the European Court of Justice in overseeing EU citizens’ rights in the UK after Brexit.
The prime minister is expected back in Brussels for further talks before the end of the week.