There is a "narrow path" to a Brexit deal this week, but the two sides have to agree the details by the end of Tuesday, the EU's negotiator Michel Barnier has said.
Mr Barnier said it was "time to turn good intentions into legal text" if EU leaders were to back the terms of the UK's exit at a summit on Thursday.
Irish Prime Minister Leo Varadkar said talks were "moving in the right direction" but gaps between the sides remained.
The UK is due to leave the EU at 2300 GMT on 31 October.
UK Prime Minister Boris Johnson has spoken to France's Emmanuel Macron and the BBC understands the two men agreed there was "positive momentum" but "many hurdles" left to overcome.
BBC political editor, Laura Kuenssberg, said she understood the two leaders had agreed on the need to avoid any further delays if possible.
But Brussels correspondent Adam Fleming said the widely-held view there was that the UK was unlikely to be leaving on 31 October, and the question was whether an extension could be short in order to iron out some small issues, or had to be much longer to deal with bigger problems.
Mr Johnson has repeatedly said Britain will leave the EU by the deadline, but he is compelled, by law, to request an extension if no deal is in place by the end of Saturday.
After updating EU ministers, Mr Barnier signalled that he expected the UK to share the legal text of any proposed changes to the withdrawal agreement - previously rejected three times by MPs - within hours.
This is designed to give enough time for European capitals to study the precise wording ahead of the two-day summit of EU leaders starting on Thursday.
He said there was a "narrow path" to be trod between the EU's objective of protecting the single market and Mr Johnson's goal of keeping Northern Ireland in the UK's customs territory.
While there had been progress in strengthening the role of Northern Ireland's political institutions in agreeing new regulatory arrangements, Mr Barnier said there was still a big disagreement about the inclusion of so-called "level playing field" provisions in the political declaration sketching out the two sides' future trade relationship.
These provisions would limit the UK's ability to diverge from the EU across a whole range of areas, including competition policy, employment rights, environmental standards and state aid.
The UK says loosening these conditions is vital if it is to have an independent trade policy, but the EU says the UK cannot have privileged access to the single market market without following its rules as this would give it an unfair advantage.
Asked whether it recognised talk of an EU deadline later on Tuesday, No 10 said Mr Johnson was "aware of the time restraints" and the UK was working hard to secure a deal "as soon as possible".
Mr Johnson is trying to hold together a coalition of Conservative Brexiteers and Democratic Unionists in support of his proposed alternative to the Irish backstop - the arrangement designed to keep an open border in Ireland.
The DUP leader, Arlene Foster, had more than an hour of talks in Downing Street on Monday night.
And on Tuesday afternoon, members of the pro-Brexit European Research Group attended a meeting at No 10.