Theresa May has been told that the DUP leader, Arlene Foster, is now “ready” to trigger a no-deal Brexit and regards this as the “likeliest outcome” following a “hostile and difficult” exchange with the EU’s chief negotiator, an explosive set of leaked government emails reveal.

Meanwhile, the former Brexit secretary David Davis has said May’s Brexit plans are “completely unacceptable” and urged cabinet ministers to “exert their collective authority”.

With continued speculation that Leave-supporting cabinet ministers could resign if May presses ahead with the plan to keep Britain in a customs union for the sake of an open Irish border, David Davis said: “This is one of the most fundamental decisions that government has taken in modern times.” Writing in the Sunday Times, he said: “It is time for the cabinet to exert their collective authority. This week the authority of our constitution is on the line.”

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 the DUP leader 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.

News of the emails comes amid new infighting in May’s cabinet over Brexit. The prime minister will make a final appeal for ministerial unity on Tuesday at a cabinet meeting, amid rumours that more senior ministers could soon follow Boris Johnson and David Davis out of the government.

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

The ministers say that if this were to be the case, the UK would never fully leave the EU. One source said eight or nine cabinet ministers had serious concerns.

The tensions have arisen because the EU has made it clear it will refuse to sign an overall withdrawal agreement – paving the way for a transition deal and the opening of talks on an EU-UK trade deal – if the UK insists on specifying a date by which customs “backstop” arrangements designed to prevent a hard border in Northern Ireland must end.

Eurosceptic ministers have insisted, however, that a date must be specified in any deal, to reassure people that the UK is serious about leaving EU structures. The Observer understands that the prime minister will try to get agreement for a compromise position, under which the UK’s intention to leave EU customs arrangements as soon as possible will be spelled out without naming a specific date.

Government sources said rumours that the Brexit secretary, Dominic Raab, might quit were overblown. May will attempt to finalise the terms of a withdrawal deal at a summit in Brussels beginning on Wednesday evening.

Meanwhile the difficulties the prime minister will have in pushing any deal she may secure through parliament are highlighted by new analysis that suggests she will have to rely on the support of more than a dozen Labour MPs to avoid her Brexit deal being torpedoed. According to researchers who examined sources including public statements, interviews and voting records of every MP in the House of Commons, May will need at least 14 Labour MPs to support her Brexit plans if they are to be passed in a vote expected to take place at the end of the year.

That rises to at least 24 should the DUP end its support for the government, were May to sign up to a Brexit withdrawal agreement that treats Northern Ireland differently from mainland Britain.

The stark research by the Edelman public affairs consultancy found that while May needs 320 votes to secure a majority, she can rely on the support of only 277 of the 314 Tory MPs and two former Tories who now sit as independents.

It predicts that 18 of the 37 hardline Tory Brexiters are “potentially persuadable”.

However, even with them on side, along with the DUP and another independent Northern Irish unionist MP, May is still 14 votes short of a majority.

Pawel Swidlicki, the Brexit analyst who led the Edelman research, said: “The good news for the PM is that she has a narrow path towards a majority. The bad news is that this not only requires reconciling the DUP with the backstop, but also winning over a raft of strongly pro-Brexit Tory MPs and a handful of Labour rebels.”