Talks have focused on the current deal's so-called 'backstop' solution, designed to keep the Irish border open but which critics say could lock Britain into a customs union with the EU indefinitely.
Talks between British and EU officials to break the Brexit deadlock will "almost certainly" last into the weekend, Britain's attorney general said on Thursday, ahead of a crucial parliamentary vote in London next week.
With just three weeks to go until the scheduled departure date of March 29, the concern is growing about the possibility of Britain crashing out of the bloc after 46 years of membership with no deal in place.
Ahead of Tuesday's vote, the British government of Prime Minister Theresa May is seeking legally-binding changes to the agreement it struck with the EU in November but which was resoundingly rejected by parliament in January.
"These discussions are running, they're going to be resuming very shortly, they're going to be continuing almost certainly throughout the weekend," Geoffrey Cox, who is leading the talks, told British MPs.
Talks have focused on the current deal's so-called "backstop" solution, designed to keep the Irish border open but which critics say could lock Britain into a customs union with the EU indefinitely.
'Still waiting for proposal'
On a visit to Britain, France's Europe Minister Nathalie Loiseau urged British politicians to "live up to the moment" as they prepared for the vote.
"So far, we are still waiting for a proposal from London, it's really a British initiative which has to come," she said.
But she later insisted on BBC radio the EU "cannot reopen" the withdrawal agreement "because it is balanced."
"The solution is on the table. The withdrawal agreement is the best possible solution."
Cox rejected criticism that Britain had not presented viable plans to the EU, telling MPs the proposals were "detailed, coherent, careful" and "as clear as day".
No-deal risks 'bitterness'
European Commission secretary general Martin Selmayr said progress was still possible, despite the failure of talks in Brussels so far.
"These things often happen at the very last minute," he told a Brookings Institution event in Washington.
"We have to wait for the next couple of days and weeks. We have to be very patient."
Loiseau said the EU did not want to activate the backstop either but rejected Britain's calls for an escape mechanism to be built into it, saying "what is not possible is that one of the parties decide unilaterally to leave".
If May loses Tuesday's vote, MPs will then vote on Wednesday on whether to proceed without a deal.
If MPs reject that outcome, they would then vote on Thursday on whether to ask the EU for a delay.
The request for a delay would have to be accepted unanimously by all member states and Britain would have to leave the EU on March 29 if it is rejected.
Loiseau said France had been "preparing for all scenarios" and that no deal "will not be a disaster for France or the European Union", although was concerned it could trigger cross-Channel animosity.
"I'm worried that... there would be bitterness between nations and there would be British-bashing in France and French-bashing or European-bashing in London," she told the Guardian newspaper.
"Why would be there be an extension without a reason?" Loiseau said.
"We have been in discussions for quite a long time now. There needs to be something specific to justify an extension," she added, pointing out that was the position of French President Emmanuel Macron and German Chancellor Angela Merkel.