Brexit negotiators are meeting now to try and resolve “big issues” ahead of a crunch summit of EU leaders this week.
The talks between the UK’s Dominic Raab and the EU’s Michel Barnier are focusing on Northern Ireland and future economic relations.
It comes as domestic political pressure on Theresa May increases.
Scottish Secretary David Mundell and Scots Tory leader Ruth Davidson say they will not accept Northern Ireland being treated differently.
In a letter to the prime minister, they said the integrity of the UK “remains the single most important issue for us” and cannot be undermined by any withdrawal agreement with the EU.
They indicate they will not tolerate a situation in which Northern Ireland remained in the customs union and single market, while the rest of the UK was outside it, or the introduction of customs and regulatory checks on goods moving between Northern Ireland and Britain that did not exist elsewhere within the UK.
A source close to Ms Davidson said the issue was a “red line” for her, while a source close to Mr Mundell told the BBC: “If you find yourself not agreeing with government policy” resigning would be the “logical outcome”.
It follows reports that other top ministers are considering their positions over the weekend ahead of a meeting of the cabinet on Tuesday at which ministers could be asked to give their consent to an agreement.
The Raab-Barnier meeting comes amid conflicting signals as to whether the two sides are nearing a deal on the terms of the UK’s exit next March as well as a framework of future economic and security co-operation.
Air of drama
By the BBC’s Brussels reporter Adam Fleming
It’s not clear for how long the two will meet, or whether the Brexit secretary will stay in Brussels overnight, although there are indications this might just be a day trip.
While Mr Raab’s weekend visit has an air of drama, it’s standard practice in the talks for civil servants to hand over to politicians at key points like this.
The government statement suggests the Northern Irish backstop is not the only outstanding issue.
At the end of last week, the unresolved areas were the governance of the Withdrawal Agreement (solving disputes that arise from the treaty) and the protection of regional products.
The EU’s aim is to have the entire Withdrawal Agreement agreed in principle by Wednesday’s summit of EU leaders.
Michel Barnier has said there should be “decisive progress” before work begins on drafting the political declaration on the future relationship. Diplomats in Brussels said they would be prepared to approve the Withdrawal Agreement if there were still small outstanding issues.
The EU has said there needs to be “decisive progress” on Northern Ireland and other issues in the run-up to the meeting of EU leaders, which starts on Wednesday.
If this happens, the EU could then hold a follow-up event in mid-November, at which any deal would be expected to be finally approved.
A Department for Exiting the European Union spokesman said: “With several big issues still to resolve, including the Northern Ireland backstop, it was jointly agreed that face-to-face talks were necessary.”
The issue of the border between Northern Ireland and the Republic of Ireland, which will become the UK’s border with the EU, is one of the last remaining obstacles to achieving a divorce deal with Brussels.
Wrangling is continuing over the nature of a “backstop” to keep the border open if a wider UK-EU trade arrangement cannot resolve it.
The EU’s version, which would see just Northern Ireland remain aligned with Brussels’ rules, has been called unacceptable by Mrs May and her Democratic Unionist allies.
And many Conservative MPs are unhappy with the UK government’s proposed alternative, which would see the UK temporarily remain in a customs union until the Irish border question is resolved, either through technological solutions or as part of a wider trade agreement.
Brexiteers fear this will leave the UK in indefinite limbo, bound by the EU’s rules and limited in the trade deals it can negotiate with other countries.
Writing in the Sunday Times, former Brexit Secretary David Davis urged ministers to “exert their collective authority” and reject the plans.
But Health Secretary Matt Hancock urged ministers to unite behind the prime minister, insisting there were “different ways” to ensure any customs commitments were “credibly time-limited”.
Asked whether any deal would include a date at which the UK would no longer be bound by the rules of the customs union, he told the BBC’s Andrew Marr Show: “I certainly hope so.”
There was “absolutely no reason” for cabinet ministers to quit over the issue, he insisted.
On Saturday evening, German newspaper Suddeutsche Zeitung reported a deal had already been reached between Mrs May and the EU, and would be announced on Monday.
But a No 10 source told the BBC the report was “100%, categorically untrue”.