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 has said.
Michel 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.
As talks intensify, Boris Johnson has spoken to France’s Emmanuel Macron.
The BBC understands the two men agreed there was “positive momentum” but there were “many hurdles” left to overcome.
Our 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.
The PM has repeatedly said the UK 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.
The Irish government has also said “big steps are needed” on Tuesday “to build on progress that has been slow”.
Tánaiste (deputy prime minister) Simon Coveney said Mr Barnier had made it clear there would be no haggling over the details of the text at the summit and the document had to be legally “watertight” and politically acceptable.
“If there is going to be a deal at this summit the task force and the British team need to finalise that text and today is a key day in terms of being able to do that in time.”
But he also suggested the summit was not the final opportunity for a deal and the process could continue into the following week.
The UK’s Brexit Secretary Stephen Barclay said both sides needed to be given the “space” to work through their detailed discussions and a “deal was still possible”.
The two-day EU summit is crucial because, under the Benn Act, the PM must get a new deal approved by MPs by Saturday if he is to avoid asking for a delay.
Labour has threatened court action to force the PM to obey the legislation, amid speculation the PM could seek to sidestep it somehow.
Regardless of what happens in Brussels, a showdown is anticipated in an emergency sitting of Parliament on Saturday – the first in 37 years. MPs will be able to back or reject any deal presented to them, and there will be discussions on what to do next.
Ex-Foreign Secretary Jeremy Hunt said while he hoped an agreement could be reached and the necessary legislation passed to implement it by 31 October, it would be “very difficult” given how little time was left.
He told BBC Radio 4’s Today that an extension of a “few days” might be acceptable to many Conservatives as it would make clear “the end was in sight”.
But Leader of the House of Commons Jacob Rees-Mogg said legislation required to ratify any agreement would be “relatively easy to pass” if MPs agreed to a deal on Saturday.
He suggested the mood was changing in Parliament and after more than three years of disagreement and rancour “everyone is desperate to finish this”.
“The votes are now there for a deal,” he insisted. “There is just a mood in the country that we want to get on with this and politicians have to be sensitive to that mood.”
Timeline: What’s happening ahead of Brexit deadline?
Thursday 17 October – Crucial two-day summit of EU leaders begins in Brussels. This is the last such meeting currently scheduled before the Brexit deadline.
Saturday 19 October – Special sitting of Parliament and the date by which the PM must ask the EU for another delay to Brexit under the Benn Act, if no Brexit deal has been approved by MPs and they have not agreed to the UK leaving with no-deal.
Thursday 31 October – Date by which the UK is currently due to leave the EU.