"The next they would hear from us would be after we leave," said one Tory insider, suggesting that Mr Johnson would boycott an EU summit on October 17-18 and make preparations for a no deal exit.
Although Mr Johnson says his plan is a "broad landing zone", he is refusing to budge on its key principles: keeping Northern Ireland within the UK customs area and ensuring that Stormont gives its consent to the region remaining part of the EU regulatory area.
Dominic Cummings, one of Mr Johnson's key aides, told Tory advisers that the government would not fundamentally change its proposals, suggesting that talks could be heading for deadlock.
However if negotiations do break down, a law is in place in Britain to stop a no-deal exit and some EU diplomats are calculating it would be better to resume talks on Brexit once there has been a UK general election.
With the impasse deepening, Brussels officials are writing off the likelihood of a deal before October 31 and are instead planning for the UK to request a negotiating extension.
Talks hit a new low on Friday after David Frost, the UK's main Brexit negotiator, told the EU that the UK would not make any "fundamental changes unless we are in a give-and-take relationship," an EU official told the Financial Times.
The UK has been told it needs to substantially revamp a 40-page paper intended to replace the backstop that is designed to avoid a hard border between Northern Ireland and the Republic.
Brussels has said it cannot accept customs checks being carried out on the island — the details of which the government wants to pin down during a 14-month transition period.
Since the UK sent its confidential paper on Wednesday last week, Brussels has found further fault in the alternative backstop.
Mr Frost told his counterparts that the UK wanted to scrap Brussels state aid rules for Northern Ireland, limit the role of the European Court of Justice in enforcing a post-Brexit deal, and was demanding access to EU goods databases after its departure. Brussels has rejected the demands.
Mr Frost is due to return to Brussels on Monday but EU capitals have said the bloc will not enter intensive "tunnel" negotiations intended to find a solution before the October summit.
"We want a deal and talks continue on Monday on the basis of our offer," said a spokesman for the UK government.
Many member states are already looking beyond the negotiations leading up to the summit and are asking whether the current talks could provide a platform for discussions after a UK election.
French officials have said they would not be rushed into accepting a poor offer just to get a deal and want assurances that the UK will not pursue disruptive tax policies or industrial standards.
Written by: Mehreen Khan and Sam Fleming in Brussels and George Parker in London
© Financial Times 2019