Donald Tusk and Theresa May are at odds over how long Brexit should be delayed for.
The EU Council president is formally telling officials from member countries to endorse a long extension until 31 March 2020.
But in a letter to Mr Tusk, the prime minister asked to delay Brexit until 30 June 2019 so British MPs could agree a withdrawal deal.
The EU Council president told officials that the only "reasonable" way out of the Brexit deadlock in parliament is to allow another year for talks.
However, the bloc could offer a clause to allow the UK to leave early if a deal is struck.
Speaking to Sky News, an EU source said Mr Tusk told officials: "The only reasonable way out would be a long but flexible extension. I would call it a 'flextension'.
"How would it work in practice? We could give the UK a year-long extension, automatically terminated once the Withdrawal Agreement has been accepted and ratified by the House of Commons.
"And even if this were not possible, then the UK would still have enough time to rethink its Brexit strategy. Short extension if possible and a long one if necessary.
"It seems to be a good scenario for both sides, as it gives the UK all the necessary flexibility, while avoiding the need to meet every few weeks to further discuss Brexit extensions."
Requesting a much shorter extension until the end of June, Mrs May wrote: "If the parties are able to ratify before this date, the government proposes that the period should be terminated early.
"The government will want to agree a timetable for ratification that allows the United Kingdom to withdrawal from the European Union before 23 May 2019 and therefore cancel the European Parliament elections, but will continue to make responsible preparations to hold the elections should this not prove possible.
"It is frustrating that we have not yet brought this process to a successful and orderly conclusion."
The prime minister had to set out future plans to the EU by this week, under the terms given by the bloc for the first Brexit extension, which is set to end at 11pm on 12 April.
Labour and the Conservatives will today continue their cross-party talks in the hope of coming to a compromise that will pass with a majority in the House of Commons.
Thursday's technical talks were "productive" according to a statement released after four and a half hours of negotiations between the party's front benches.
Tom Watson, Labour deputy leader, told Sky News: "Parliament is very unhappy with what's going on and the public is rightly concerned.
"We've had 1,000 days of Brexit chaos.
"The sad truth is we were going to end up with crisis talks after the election when Theresa May returned without a majority, but she could have reached out earlier and we might have had a compromise a year or 18 months ago."
Mr Watson said he was a "reluctant" convert to the idea of a confirmatory ballot, saying it had been parliament's job to deliver on Brexit, but that it had failed.
He said: "The idea of a ballot is to say 'sorry, we can't solve this, take the deal away, look at it and if it works for you, your families and your communities, then fine'."
Debates in the House of Lords over a bill brought by Sir Oliver Letwin and Yvette Cooper, which would enshrine in law the opposition to a no-deal Brexit meaning the UK cannot crash out without a deal lasted until nearly 11pm, but the final stages will not go through this week.
The bill, once law, would send Mrs May to Brussels to request another extension at the EU summit on 10 April.
If the EU required a different extension, she would have to come back to the Commons to have that approved.
Attorney General Geoffrey Cox said the bill would leave Mrs May with little room to manoeuvre.
Sky News
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