As negotiations continue before Wednesday’s meeting, the prime minister is struggling to appease her allies in the Democratic Unionist party and Brexiters in the Conservative party over a plan that could keep the UK in a customs union to avoid a hard border with Ireland.

Writing in the Sunday Times, Davis said the plan was “completely unacceptable” and urged Cabinet ministers to “exert their collective authority”.

“This is one of the most fundamental decisions that government has taken in modern times. It is time for the Cabinet to exert their collective authority. This week the authority of our constitution is on the line,” he wrote.

The issue of the border between Northern Ireland and the Republic is one of the last remaining obstacles to achieving a deal with Brussels, with wrangling continuing over the nature of a “backstop” to keep the frontier open if a wider UK-EU trade arrangement cannot resolve it.

The EU’s proposal would see Northern Ireland remain aligned with Brussels’ rules, but has been called unacceptable by May and the DUP.

May’s counter-proposal is for a “temporary customs arrangement” for the whole of the UK, but Tory Brexiters are suspicious this could turn into a permanent situation.

Davis 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”.

The DUP leader, Arlene Foster, emerged from a meeting last week in Brussels with Michel Barnier, the French official leading the EU’s negotiating team, convinced that the prospects for a Brexit deal were fading so fast that, given Brussels’ stance on Northern Ireland, an agreement had become the least likely outcome. Senior government advisers were swiftly informed that Foster was “ready” for the UK to crash out of the EU without a deal.

According to a private email exchange between senior UK officials, seen by the Observer, Foster had expressed deep disappointment about her meeting with Barnier, and outlined her wider thinking, during a dinner with the leader of the Conservative MEPs, Ashley Fox.

“She described Barnier as being difficult and hostile in her meeting today …” the leaked email from an adviser involved in the Brexit talks says. “AF [Arlene Foster] said the DUP were ready for a no-deal scenario, which she now believed was the likeliest one.”

The official added in his email, circulated at the highest levels, that it was not clear whether Foster was seeking to threaten the government or simply inform it of her plans.

Last week, the DUP, whose 10 MPs prop up May’s government, made it clear they would be prepared to vote down the budget this month if the government pressed ahead with a Brexit deal that tied Northern Ireland closer to the EU than the rest of the UK. Losing a budget vote would plunge the government into crisis.

The prime minister will call for ministerial unity on Tuesday at a cabinet meeting. Reports claim that more senior ministers could soon follow Boris Johnson and David Davis out of government.

Several cabinet ministers, including the leader of the Commons, Andrea Leadsom, and the work and pensions secretary, Esther McVey, were this weekend said to be “seriously considering their positions” because they believe May’s negotiations would leave open the possibility of the UK remaining in the EU customs union for good.

The Sunday Times said at least nine ministers want May to change course when the Cabinet meets on Tuesday.

The Northern Irish situation is politically problematic for the prime minister because her minority administration depends on the votes of the 10 DUP MPs.

A further demonstration of Tory resistance to May’s plans came from the MP Anne-Marie Trevelyan, who wrote in the Sunday Telegraph that “the idea of remaining in the customs union – even it if is called a ‘temporary customs arrangement’ – after the end of the transition period, into the 2020s, means simply delaying Brexit and causing the 17.4 million people who voted for it to lose faith in our democracy”.

Jacob Rees-Mogg, chairman of the European Research Group of pro-Brexit MPs, warned that when it came to the backstop proposal “temporary means eternal”.

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