European officials insist the deadline of next Friday is not an ultimatum to Prime Minister Boris Johnson's government but a simple fact of the Brussels diplomatic calendar.
British and EU negotiators will need a jointly-agreed legal text by then to show to EU capitals ahead of the October 17-18 European Council summit.
Most European diplomats are sceptical that such a detailed deal can be drawn up in time, despite intense ongoing technical discussions with the British side.
And if the deadline is not met, Johnson will either have to request a Brexit extension or see Britain quit the bloc without a deal.
An official at the European Council, which represents member states, confirmed that its administrative chief had warned EU ambassadors about the Friday cut-off.
"The secretary general did not impose an ultimatum but noted that we need clarity before the end of next week if the European Council is to succeed," she said.
A senior European official confirmed the timeline to AFP, adding that the draft legal text must be agreed between the UK side and EU negotiator Michel Barnier's team.
This seems a tall order, especially since European officials refuse to characterise the ongoing talks as "negotiations" at all, describing them as a technical dialogue.
"There has not been enough progress to begin a negotiation," a European Commission official said.
Johnson's diplomatic "sherpa" David Frost is in Brussels for talks with Barnier's team but no date has been set for his Brexit minister Stephen Barclay to come.
Frost brought with him a British proposal that would see measures put in place to ease trade on the Irish border when Great Britain and Northern Ireland leave the EU customs union.
This would mean removing the so-called "Irish backstop" which under a previous rejected withdrawal agreement would have seen the UK remain in the customs bloc, a key British demand.
But Brussels and Dublin are deeply sceptical of the idea, and senior European officials and diplomats warned that deep disagreements remain over Johnson's backstop alternatives.
In particular, they are not prepared to sign a deal that would require Britain to invent and deploy technological means to monitor border trade without customs posts during a subsequent transition period.
And they oppose giving the currently suspended North Ireland provincial assembly at Stormont a vote every four years on whether to remain in the post-Brexit customs arrangements.
"Yes, serious issues outstanding regarding customs and Stormont," an EU source said, warning that these would have to be resolved by Friday "if you want a deal by EUCO next week."
When Johnson sent his proposal for a revised deal this week, Number Ten officials initially briefed reporters that it was a "take it or leave it" final offer.
But the British leader and the UK officials have since moderated their tone, and Johnson now describes the plan a broad "landing zone" and spur to further talks.
"We want to get this done as soon as possible," a UK government spokesman told AFP on Friday.
"We have made a significant move this week. The EU now needs to engage with the UK's proposals if they want to get a deal."
If Johnson fails to reach a deal by October 19, he must ask Brussels to delay Brexit under a law passed by British MPs to try to prevent a "no-deal" divorce.
He insists, however, that he will not do this and Britain will leave on October 31 come what may.