Prime Minister Theresa May is to travel to Brussels on Wednesday for the start of the summit, when both sides want to sign off on a draft withdrawal agreement to set out the terms for Britain's divorce from the EU.
European Union flags are seen outside the EU Commission headquarters in Brussels, prior to a meeting between Britain's Secretary of State for Exiting the EU Dominic Raab and EU's chief Brexit negotiator Michel Barnier, Belgium October 14, 2018.
It comes as domestic political pressure on Theresa May increases. Penny Mordaunt, the global development secretary, and Esther McVey, the work and pensions secretary, and Andrea Leadsom have the most significant concerns over the plans, although they did not attend last night's meeting.
D-U-P leader, Arlene Foster, added that the European Union's proposal for a backstop arrangement will effectively keep Northern Ireland in the bloc.
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 several Eurosceptic Cabinet ministers will hold a meeting this evening to coordinate their response to Mrs May's Brexit plans.More news: Deshaun Watson Injury Keeping Bills-Texans Betting Line Off the Board
But after talks which lasted a little over an hour, it was clear that obstacles remained.
Scottish secretary David Mundell and Ruth Davidson, leader of the Scottish Tories, have also made clear they would resign if Northern Ireland faces new controls that separate it from the rest of the United Kingdom - because that would fuel the case for Scottish independence.
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.
Some EU diplomats speculate that she needs to have a fight in order to get the deal she does eventually deliver through a divided British Parliament. Brussels hoped to have a provisional deal in place this month, which could be signed off by European Union leaders at their next summit in November.
If this happens, the European Union could then hold a follow-up event in mid-November, at which any deal would be expected to be finally approved.
Dominic Raab's office said in a statement that the unusual weekend meeting with the EU's Michel Barnier was necessary.More news: Bitcoin Whales Might Not Be the Ones Manipulating The Market
The Irish backstop measure, created to ensure no new border emerges between Ireland and Northern Ireland, is supposed to be temporary until a UK-EU trade deal can be agreed, and Theresa May pledged earlier this year that the arrangement would expire "at the very latest by the end of December 2021".
One of the issues that has been hampering progression of a deal has been that of the Irish border and a potential backstop - a mechanism to avoid a hard border with Northern Ireland.
The head of a Northern Irish party that props up her government also raised the stakes, saying it was ready to trigger a so-called no deal Brexit to prevent what it called "the annexation" of the British province by the EU.
However, Brexiteers like Patel and Davis suspect this could turn into a permanent situation, restricting Britain's freedom to strike future trade deals with other countries.
David Davis, who quit as Brexit secretary in July over May's broad blueprint, wrote in The Sunday Times newspaper her plans were "completely unacceptable" and urged ministers to "exert their collective authority" this week.More news: Why the Dallas Cowboys will lose big at home to the Jaguars