European Council President Donald Tusk is proposing to offer the UK a 12-month "flexible" extension to its Brexit date, according to a senior EU source.
The plan would allow the UK to leave the EU sooner if the British Parliament ratifies a deal, but it would need to be agreed by European leaders at a summit next week.
The UK is due to leave the EU on 12 April, but no withdrawal deal has yet been approved by MPs.
Downing Street said "technical" talks between Labour and the Conservatives on Thursday had been "productive" and would continue on Friday.
Prime Minister Theresa May has said a further postponement to the Brexit date is needed if the UK is to avoid leaving the EU without a deal, a scenario EU leaders and many British MPs believe would create problems for businesses and cause difficulties at ports.
However, the PM wants to keep any delay as short as possible.
To do that, she and Labour leader Jeremy Corbyn would need to agree a proposal for MPs to vote on before 10 April, when EU leaders are expected to consider any extension request at an emergency summit.
If they cannot, Mrs May has said a number of options would be put to MPs "to determine which course to pursue".
UK Attorney General Geoffrey Cox said that particular scenario would involve accepting whatever postponement the EU offered, which was likely to be "longer than just a few weeks or months".
He said that if talks failed, the delay was "likely to be a long one".
Europe's leaders have been split over whether, and how, to grant any extension.
A senior EU official told BBC that Mr Tusk "believes he's come up with an answer" after several hours of meetings in preparation for the summit.
"He believes the arrangement would suit the EU and the UK and, as one EU official put it to me, it would avoid Brussels potentially being faced with UK requests for repeated short extensions every few weeks," she said.
The EU has previously said that the UK must decide by 12 April whether it will stand candidates in May's European Parliamentary elections, or if the option of a long extension to Brexit would become impossible.
Talks between Conservative ministers and Labour lasted 4.5 hours on Thursday.
Mr Corbyn has written to his MPs saying discussions included customs arrangements, single market alignment, internal security, legal underpinning to any agreements and a "confirmatory" vote.
Shadow Treasury minister Clive Lewis said the party would not be talking to the government if a "confirmatory referendum" was not an option.
However, 25 Labour MPs - including a number representing Leave-voting seats - have written to Mr Corbyn, saying another referendum should not be included in any compromise Brexit deal.
Asked whether another referendum on any final deal was a credible option, Mr Cox said: "A good deal of persuasion might be needed to satisfy the government that a second referendum would be appropriate. But of course we will consider any suggestion that's made."
If the talks fail, the government faces an additional obstacle in the form of a backbench bill which would force the PM to seek a new delay.
Passed by MPs by one vote on Wednesday, the bill is being scrutinised by the House of Lords, who will next consider the draft legislation on Monday.
Ministers have argued it could increase "the risk of an accidental no-deal" in the event the EU agreed to an extension but argued for a different date than one specified by MPs.
That would mean Mrs May having to bring the issue back to the Commons on 11 April, when European leaders would have returned home, the prime minister's spokesman said.
After a meeting with Irish Prime Minister Leo Varadkar in Dublin, German Chancellor Angela Merkel said her country still hoped for an "orderly Brexit".
"We will do everything in order to prevent... Britain crashing out of the European Union," she said.
"But we have to do this together with Britain and with their position that they will present to us."