The EU and UK will hold further talks on changes to their Brexit deal to help Theresa May get it through the Commons.
It comes after what both sides said was a « robust but constructive » meeting between Mrs May and European Commission President Jean-Claude Juncker.
Mrs May wants to prevent the UK being « trapped » in the backstop – the plan to keep the Irish border open.
Mr Juncker ruled out legally-binding changes to the backstop clause in the withdrawal agreement.
But he said the EU would be open to adding words to the non-binding future relations document that goes with the withdrawal agreement, following their meeting.
Brexit Secretary Steve Barclay and EU negotiator Michel Barnier will hold talks in Strasbourg on Monday, as the EU and UK Brexit negotiating teams discuss proposed changes to the deal.
British sources say the talks will include discussion of the legally-binding withdrawal agreement, the BBC’s Brussels reporter Adam Fleming said.
An EU source says the further talks are an opportunity to listen to the UK’s ideas.
Mrs May and Mr Juncker will meet again before the end of February, to review progress.
The prime minister is expected to put the deal to a vote in the Commons towards the end of February.
She says the plan must change if it is to win the support of MPs who urged her to seek « alternative arrangements » when rejecting the deal last month.
In their joint statement following talks in Brussels, Mrs May and Mr Juncker said: « The prime minister described the context in the UK Parliament, and the motivation behind last week’s vote in the House of Commons seeking a legally-binding change to the terms of the backstop.
« She raised various options for dealing with these concerns in the context of the Withdrawal Agreement in line with her commitments to the Parliament. »
Mr Juncker « underlined that the EU27 will not reopen the withdrawal agreement » but « expressed his openness » to amending the future relationship to be « more ambitious in terms of content and speed ».
Mrs May also held a meeting with European Parliament President Antonio Tajani and the Parliament’s Brexit co-ordinator Guy Verhofstadt.
Mr Verhofstadt said Mrs May had « assured us that there would be a backstop » in the withdrawal agreement.
She will later hold talks with European Council President Donald Tusk, who sparked controversy on Wednesday by declaring there was a « special place in hell » for those who campaigned for Brexit without a plan to deliver it safely.
MPs who backed Leave in the 2016 referendum reacted with anger to the comments, accusing Mr Tusk of « arrogance ».
David Lidington, the minister seen as Mrs May’s second-in-command, said Mr Tusk’s comment « wasn’t the most brilliant diplomacy in the world » but would « detract from what I expect to be a courteous and sensible grown-up discussion ».
Labour’s fresh offer to May
Jeremy Corbyn has written to the prime minister setting out his party’s price for supporting a Brexit deal and to offer talks to secure « a sensible agreement that can win the support of Parliament and bring the country together ».
The Labour leader’s five demands include a « permanent and comprehensive UK-wide customs union » aligned with the EU’s customs rules but with an agreement « that includes a UK say on future EU trade deals ».
Mr Corbyn also wants close alignment with the EU single market and « dynamic alignment on rights and protections » for workers so that UK standards do not fall behind those of the EU.
He also proposes participation in EU agencies and funding programmes, and agreements on security and keeping access to the European Arrest Warrant.
The letter does not mention previous demands that any deal must deliver the « exact same benefits » that membership of the single market and customs union currently does – effectively scrapping the party’s « six tests » that had been its Brexit policy.
The BBC’s Political Editor Laura Kuenssberg said Labour’s shadow Brexit secretary Sir Keir Starmer would be meeting cabinet office minister David Lidington to discuss the proposals.
But the move was met with dismay by Labour members of the People’s Vote campaign for another EU referendum, who accused Mr Corbyn of going back on his commitment, made at the party’s conference, to back a public vote if he can’t force a general election.
Labour MP Chris Leslie tweeted: « Seriously? Offering to help Tory Govt enable Brexit? It’s not just Labour’s conference policy in the bin.
« When the jobs go & revenues for services dry up as a result – Labour’s leadership will have ZERO right to complain: they share responsibility. »
Labour MP Owen Smith, who made a failed leadership bid in 2016, has told the BBC he and others were thinking of quitting the party over Mr Corbyn’s Brexit stance.
But Labour’s Stephen Kinnock, who backs the « Norway Plus » model of a close economic partnership with the EU, welcomed Mr Corbyn’s letter, tweeting: « This can break the deadlock. »
Mr Lidington said he and Brexit Secretary Steve Barclay would be willing to discuss Labour’s proposals with Sir Keir and other frontbenchers.
But he said Labour’s call to have a say in trade deals while being in a customs union with the EU was « wishful thinking », because Brussels had ruled it out.
Analysis: What awaits Theresa May in Brussels?
By BBC Europe editor, Katya Adler
There’s pretty much zero expectation that any real progress will be made on Thursday when Theresa May comes back to Brussels looking for changes to the backstop.
Mr Tusk is not alone in Europe in his frustration at leading Brexiteers’ unrealistic promises…
EU leaders are irritated too that – as they see it – the UK voted for Brexit but keeps looking to Brussels to come up with ways to make its exit workable and painless.
But most senior European politicians are keeping those thoughts quiet – in public.
Considering the tortuous political dance Theresa May is trying to pull off in Westminster, they realise outspokenly critical EU opinions may not be helpful if, in the end, they want to get this Brexit deal done with the UK.
- Read Katya’s analysis in full
Where are we in the Brexit process?
The UK is due to leave the EU at 23:00 GMT on Friday 29 March, when the two-year time limit on withdrawal negotiations enforced by the Article 50 process expires.
In January, MPs overwhelmingly rejected the withdrawal deal that the government had negotiated with the EU, backing an amendment for the government to seek « alternative arrangements » to the backstop.
The backstop is an « insurance policy » designed to avoid « under all circumstances » the return of customs checkpoints between Northern Ireland and the Irish Republic after Brexit.
Many fear creating physical infrastructure along the border could threaten the peace process.
But the Democratic Unionist Party and Brexiteers believe the proposed temporary single customs arrangement could threaten the integrity of the UK, leaving it bound by EU rules if no trade deal is agreed.