Theresa May will make a last-ditch attempt to persuade the EU to give her a better Brexit deal on Friday, as she struggles to hold her crumbling government together following a day of cabinet embarrassments in Westminster.

The prime minister will plead with EU leaders to offer further concessions, as it became clear that talks in Brussels have stalled and hardline Eurosceptics in her party are likely to vote down the deal for a second time in parliament next week.

Senior Tory critics of May expressed astonishment that her strategy was a refusal to change course in the face of defeat, with one cabinet source saying No 10 realised it was about to lose the meaningful vote but seemed unable to make a coherent case to MPs why they should vote for it.

Instead, May will turn her fire at the EU in a speech delivered from the Brexit stronghold of Grimbsy in Lincolnshire, saying: “Just as MPs will face a big choice next week, the EU has to make a choice too.

“It is in the European interest for the UK to leave with a deal. We are working with them but the decisions that the European Union makes over the next few days will have a big impact on the outcome of the vote.”

The speech will take place after a series of unforced errors by three cabinet ministers, forcing May’s chief spokeswoman to insist the prime minister still had confidence in them all.

As cabinet discipline appeared to break down, Karen Bradley, the Northern Ireland secretary, had to issue a grovelling apology for saying that deaths caused by police and soldiers during the Troubles were “not crimes”.

Then Amber Rudd, the work and pensions secretary, had to say sorry for referring to the shadow home secretary, Diane Abbott, as a “coloured woman”, and Andrea Leadsom, the leader of the House of Commons, suggested it should be for the Foreign Office to hold a debate on allegations of Islamophobia in the UK.

One Conservative former cabinet minister said it was “like the last days of Rome” and it was difficult to see how May would last many more weeks, even if her Brexit deal scraped through under heavy pressure.

The consensus among dozens of Tory MPs and ministers spoken to by the Guardian was that the withdrawal deal would not pass, as it appeared unlikely that Geoffrey Cox, the attorney general, would secure a legally binding agreement from the EU that the Irish backstop was temporary. Without such an assurance, the hardline Eurosceptics in the European Research Group (ERG) and the Democratic Unionist party will have no reason to change their vote, as they will not countenance the idea that the UK could be bound into a permanent customs union with the EU.

Given the likelihood of defeat, there was heavy speculation in Westminster that May will end up pulling the votes she had promised MPs on whether to rule out a no-deal Brexit and a second on whether to extend article 50.

A senior government adviser said she could announce straight after losing the meaningful vote that she would seek an extension from the EU, knowing that she would only get either a short technical extension to enable the passing of her withdrawal deal or a lengthy 21-month one enabling a full rethink – dubbed a Brexit reset. The prime minister would then return to the House of Commons for a third time to put those proposals to a vote.

May has a serious problem with holding a vote on whether parliament should rule out no deal, as it is unclear which side of the argument she would be on and would risk resignations of cabinet ministers from each side of the Brexit debate if she whipped in either direction. It is understood a delegation of cabinet ministers met her chief of staff on Thursday to argue for a free vote.

But there was a growing feeling that she may attempt to avoid the problem altogether by moving first and offering to secure an extension to article 50. “It will get very, very nasty next week,” said one source close to a cabinet minister. “I think No 10 promised the votes never really thinking they were going to have to hold them because they were naively convinced the deal would go through.”

With May insisting Brussels must move first, Philip Hammond, the chancellor, took a different approach on Thursday, arguing that hardline Eurosceptics should realise their refusal to back the government’s deal was risking a softer Brexit. His comments were a strong hint that May would have to reach out to Labour and try to agree a Brexit involving a permanent customs union if the deal is not voted through in its current form.

“The Labour party has been talking for a long time about the idea of a customs union grafted onto the PM’s deal. Those of my colleagues who feel very strongly against that proposal need to think very hard about the implications of voting against the prime minister’s deal next Tuesday, because we will then be in unknown territory where a consensus will have to be forged across the House of Commons and that will inevitably mean compromises being made,” he told BBC Radio 4’s Today programme.

Amid an absence of ideas about how to break the stalemate, former prime minister Gordon Brown joined a group of business leaders in calling for a one-year extension to article 50 in order to set up region by region public hearings on which Brexit option is preferred.

Responding to May’s speech, Keir Starmer, Labour’s shadow Brexit secretary, said it was “becoming increasingly clear that Theresa May will not be able to deliver the changes she promised to her failed Brexit deal [and] this speech looks set to be an admission of failure.”

There were reports from Brussels on Thursday night that the EU was open to making an offer on bolstering the review system for the Irish backstop but it was not clear this would be enough for Cox to change his legal advice that the mechanism could bind the UK into a permanent customs union.

Earlier, in the House of Commons, Cox rebuffed the idea from France’s Europe minister, Nathalie Loiseau, that the UK should make a new substantial proposal to break the impasse.

“I am surprised to hear the comments that have emerged over the last 24 hours that the proposals are not clear. They are as clear as day and we are continuing to discuss them,” Cox said.

He refused to divulge the details of what he said were “detailed, coherent, careful proposals”. But it is understood Cox has been pushing for the arbitration panel to be able to decide whether the EU has made reasonable efforts to negotiate alternatives to the backstop, opening up the opportunity for the UK to exit the arrangement in part or whole.

Sabine Weyand, the EU’s deputy chief negotiator, gave what sources described as a “gloomy” briefing to ambassadors on Wednesday night about Cox’s meetings this week, and Downing Street has admitted talks have been difficult.