The unscheduled talks in Brussels came amid speculation officials working on the negotiations had reached a deal.
With the negotiations coming to a head, the central focus of the discussions is thought to have been the issue of the Northern Ireland "backstop" meant to ensure there is not return of a "hard border" with the Republic.
Downing Street has been insistent that it does not expect the backstop be needed, as it will be possible to agree a deal that maintains frictionless trade across the Irish border by the end of the transition period in December 2020.
A UK Government spokesman said: "Brexit secretary Dominic Raab will be in Brussels this afternoon to meet with Michel Barnier".
But the Guardian and Daily Telegraph reported that the option of a potential extension was being considered by negotiators in Brussels to allow extra time to drawn up a deal on the future UK-EU relationship - and avoid the need to use a controversial "backstop" arrangement to avoid a hard border between Ireland and Northern Ireland.
And Tory MP Nadine Dorries suggested Mr Davis should be installed as interim leader, claiming that was the only way to secure the kind of free-trade deal Brexit demanded by Eurosceptics.
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Irish Foreign Minister Simon Coveney insisted the backstop should be able to remain in place "unless and until something better comes along" and told ITV News it was a "deal breaker".
The deal believed to be on the table involves keeping the whole United Kingdom in an "arrangement" that effectively preserves the existing EU customs union, ensuring the goods continue to move freely over the Irish land border regardless of the future trade relationship between London and Brussels.
Foreign Secretary Jeremy Hunt, who was hosting eight European Union counterparts at his Chevening country retreat, said "there is no-one who is going to be able to negotiate the right deal for Britain better than Theresa May", adding "she is battling for Britain".
The deadline for a Brexit deal is fast approaching.
Writing in the Sunday Times, Davis said the plan was "completely unacceptable" and urged Cabinet ministers to "exert their collective authority".
Whether it is possible is likely to become clear next week, when the prime minister heads to Brussels for what could be a potentially historic summit with other European Union leaders.
Pik Botha has died at the age of 86
The reduction in regional tensions was followed by the 1990 release of Mandela, who had spent 27 years in apartheid prisons. Botha was appointed minister of mineral of energy affairs in 1994 and resigned from that post in May 1996.
The Guardian said House of Commons leader Andrea Leadsom was also concerned about May's reported plan to accept keeping Britain in the EU customs union for an indefinite period after it withdraws.
Mrs May's own position also appeared in jeopardy, with as many as 44 letters demanding a vote of no confidence reportedly submitted to the Conservative 1922 Committee - just four short of the number required to trigger a ballot.
The prime minister is rumoured to be making a statement later on Friday to clarify her position.
Davis also pressed May to abandon her Brexit proposal, which involves staying in a free trade zone with the European Union for goods.
Mr Davis is understood to be prepared to run if there is a leadership contest.
Mr Barnier has been touring European Union capitals for months and visited Warsaw today as Brexit talks have entered a crunch period. Many, notably French President Emmanuel Macron, made clear at a summit with May last month that they would agree to meet only if she could show she had come sufficiently close to a deal to make it worthwhile.
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Asked by the BBC to offer her backing to Mrs May's plan, Ms McVey sidestepped the questions.