The UK could remain tied to Brussels' rules beyond the end of 2020 to give negotiators more time to finalise a trade deal, according to reports.
Northern Ireland's Democratic Unionist Party on Thursday threatened to withdraw its support for British Prime Minister Theresa May if she compromises with Brussels in a move that could bring down the government and trigger an election.
Writing in The Sunday Times, the former minister said the government's negotiating strategy had "fundamental flaws", arising from the "unwise decision in December to accept the EU's language on dealing with the Northern Ireland border".
Scottish Brexit Secretary Michael Russell said: "The reality of a disastrous no-deal Brexit looms large in this latest guidance from the UK Government".
Allies of Mr Davis said he has been contacted by several Tory MPs urging him to run for the leadership and he is understood to be prepared to take part in a contest.
Chancellor Philip Hammond became the first senior Government figure to suggest that the backstop - created to keep the border between Northern Ireland and the Republic open in the case that no broader EU/UK trade deal is finalised - will "probably" have to come into effect for a period.
Britain and the European Union called a pause in the Brexit negotiations until leaders of member states meet in Brussels, four European Union sources told Reuters, dealing a blow to hopes that a flurry of talks could secure a draft agreement beforehand.
Mrs May's counter-proposal is for a "temporary customs arrangement" for the whole United Kingdom, but Tory Brexiteers fear this becoming an open-ended position which would prevent free trade deals with countries around the world.
The Prime Minister faces a showdown over her Chequers plan this week after negotiations stalled yesterday, with an European Union summit set to take place on Wednesday.
European Union sources said Barnier's team have offered new wording on a text on the backstop, highlighting that it may never be activated or only for a limited time, and also that Britain as a whole could stay under European Union rules for longer as it transitions out after March 2019.
Tory vice chairman James Cleverly told Sky News' Sophy Ridge on Sunday that Cabinet ministers should "use that position to put their ideas forward, I would suggest that's a much more constructive and helpful way of influencing the direction of the negotiations with the EU".
Raab's predecessor, David Davis, wrote in the Sunday Times that May's plans for continued close economic ties with the European Union even after Britain leaves the bloc is "completely unacceptable" and must be stopped by her ministers.
Theresa May now needs to sell it to her ministers at Tuesday's crunch cabinet meeting.
House of Commons leader Andrea Leadsom, International Development Secretary Penny Mordaunt and Work and Pensions Secretary Esther McVey were all poised to quit.
The warnings issued to Mrs May about the DUP's "red lines" on Brexit are created to signal Mrs Foster's tough leadership qualities to her members - but Nigel Dodds, the party's Commons leader, is increasingly being talked up as her replacement.
Tory MP Nadine Dorries publicly suggested he could be the leader to deliver the kind of Brexit sought by Eurosceptics.
In the event of a Tory leadership battle, Davis would look to position himself as the Brexiteer's natural "candidate of choice", The Mail on Sunday says. "We are not going to see Northern Ireland carved off".
"So it is probably inevitable that we will end up with a no deal scenario".
So far, May has shown little appetite to change tack, trying to persuade Conservative lawmakers and those in the opposition Labour Party to vote for any deal based on her plan.
But shadow foreign secretary Emily Thornberry told the BBC: "If she comes back with something that's just a fudge she's cooked up with Brussels ... we're not voting for something that's essentially a bridge to nowhere".