Theresa May will take the high-stakes battle to save her Brexit plan and her premiership to all four nations of the United Kingdom this week after warning MPs on Monday that failing to back the deal will take Britain “back to square one”.

EU leaders gave their formal backing to the Brexit deal at a special summit in Brussels on Sunday, after more than a year and a half of painstaking negotiations. But they were adamant that there was no plan B other than a no-deal exit if MPs were to reject it in the House of Commons in a “meaningful vote” expected to be held on 12 December.

At a press conference after Sunday’s summit, the prime minister hailed the historic moment as “the culmination of a long and difficult process”, and said she had rejected the “counsels of despair” that suggested a deal was impossible. She refused to rule out resigning if the deal is rejected, saying, “it’s not about me”.However, with a growing number of her own MPs setting their faces against it, and potential Labour rebels falling away, winning the Commons vote will be difficult.

May will gather her cabinet on Monday morning before addressing parliament. She will warn anxious MPs, “we can back this deal, deliver on the vote of the referendum and move on to building a brighter future of opportunity and prosperity for all our people.

“Or this house can choose to reject this deal and go back to square one. It would open the door to more division and more uncertainty, with all the risks that will entail.”

A bid to topple May as Conservative leader, launched by European Research Group chair Jacob Rees-Mogg, failed to gather the requisite support from MPs.

But May has said she believes “with every fibre of my being” that her deal is the right one for Britain; and after a string of cabinet resignations, including those of two Brexit secretaries, her reputation is intimately bound up with the plan.

The foreign secretary, Jeremy Hunt, conceded on Sunday that getting the deal approved by the Commons looked “challenging”.

Asked if this meant it was possible May’s government could collapse, Hunt did not dismiss the idea. He said: “It’s not possible to rule out anything, and that’s why all of us have to say: what do your constituents actually want in this situation? And we have to work out what’s in the national interest, and it’s all about the balance of risks.”

The French president, Emmanuel Macron, piled further pressure on the prime minister over the issue of fisheries. He warned that unless the UK swiftly agreed that European fleets would continue to have similar access to British seas as today, Britain would have to fall back in the long term on the customs union envisioned in the withdrawal agreement, which was designed as a last resort to solve the Irish border problem should a wider deal not be in place.

May said on Sunday she wanted to “speak directly to the British people”, and explain the benefits of the deal to them – highlighting an end to the free movement of people, “once and for all”.

She also pointed to the extra resources her government plans to devote to the NHS, which she has previously claimed – controversially – to be funded partly by a “Brexit dividend”.

May also highlighted the fact that her deal will allow Brexit to go ahead as planned in March, so the government can switch some of its focus to domestic priorities. “The British people don’t want to spend any more time arguing about Brexit,” she said.

“I will take this deal back to the House of Commons confident we have achieved the best deal available, and full of optimism about the future of our country,” she added. “In parliament and beyond it, I will make the case for this deal with all my heart, and I look forward to that campaign.”

This direct appeal to the public has sparked suspicion among some senior Labour figures that the prime minister could be preparing to fall back on the unpalatable option of a general election if her deal is rejected by MPs.

Labour leader Jeremy Corbyn said: “This is a bad deal for the country. It is the result of a miserable failure of negotiation that leaves us with the worst of all worlds. It gives us less say over our future, and puts jobs and living standards at risk. That is why Labour will oppose this deal in parliament.”

After endorsing the withdrawal agreement and political declaration following a discussion lasting less than an hour at Sunday’s summit, the EU’s leaders made a direct appeal to MPs to back their prime minister.

“I am inviting those who have to ratify this deal in the House of Commons to take this into consideration: this is the best deal possible for Britain, this is the best deal possible for Europe,” the European commission president, Jean-Claude Juncker, said. “This is the only deal possible – the only deal possible.”

EU officials said the only “plan B” currently being contemplated after “long, tortuous” negotiations were no-deal preparations. A number of leaders wished the prime minister good luck in parliament during the summit.

The president of the European council, Donald Tusk, told reporters: “Ahead of us is the difficult process of ratification, as well as further negotiations. But regardless of how it will all end, one thing is certain: we will remain friends until the end of days, and one day longer.”

On the fisheries issue, Macron said: “We, as 27, have a clear position on fair competition, on fish, on the subject of the EU’s regulatory autonomy, and that forms part of our lines for the future relationship talks.

“It is a lever, because it is in our mutual interest to have this future relationship. I can’t imagine that the desire of Theresa May or her supporters is to remain for the long term in a customs union, but to define a proper future relationship which resolves this problem.”

Juncker said Sunday was a “sad day”. Asked if she shared the sentiment, May said: “No: but I recognise that others do. I recognise that some European leaders are sad at this moment.”

However, former foreign secretary Boris Johnson, said EU leaders would welcome the deal but it was a “disaster” and a “humiliation” for the UK. In his regular Daily Telegraph column, he said: “The other EU countries have signed the deal immediately, because they know that they have us exactly where they want us. We are a satellite state – a memento mori fixed on the walls of Brussels as a ghastly gaping warning to all who try to escape.” Restating his calls for the backstop to be scrapped, the full payment of £39bn to be withheld until a trade deal is done, and intensified planning for a no-deal Brexit, Johnson said: “We cannot go on any more with this hopeless can’t-do spirit; this reflexive negativity and defeatism that so woefully underestimates the courage and creativity of the British people.”

The parliamentary arithmetic appeared yet more challenging for the prime minister on Sunday, as potential Labour rebels whom she had hoped might support deal appeared to be falling away. Lisa Nandy, the influential Wigan MP, who had previously suggested she would weigh up voting with the government, told Sky News it was “inconceivable” she could back it.

Gareth Snell, who beat former Ukip leader Paul Nuttall in Stoke Central, and had expressed concerns about the prospect of a no-deal Brexit, is also minded to obey the Labour whip and vote against, he told the Guardian.