A deep cabinet split has opened up over whether Theresa May should back a second referendum in a final attempt to end the political deadlock over Brexit, as senior Conservatives predicted on Saturday night that her blueprint for leaving the EU was heading for a crushing House of Commons defeat.

Adding to a mounting sense of constitutional crisis ahead of Tuesday’s crucial parliamentary vote, No 10 is braced for more resignations of ministers and aides who want another referendum, or who believe May’s deal fails to deliver on Brexit. Will Quince, the Colchester MP and aide to the defence secretary, Gavin Williamson, quit his post on Saturday night in protest at the Brexit deal.

Cabinet ministers have told the Observer that attempts to convince May to delay the vote to avoid one of the largest and most humiliating defeats in recent parliamentary history had not been heeded. This was despite what they saw as a clear danger that such a result could provoke a leadership challenge and split the party irrevocably. Some cabinet ministers now believe that May is so wedded to her Brexit deal that her only method of gaining approval will be through another referendum – and that the arguments for a second vote are emerging as stronger than those for a soft Brexit. The prime minister has so far refused to entertain any idea of a second public vote.

One cabinet source said it might prove to be the only way of saving May’s deal and her reputation. “She is so committed to her deal, and a second referendum could now be the only way of getting it. The polls have been remarkably stable for a while, but there does seem to be some kind of movement [to Remain], and that could well develop in the coming days and weeks.”

Another senior Tory backing a second referendum said: “There are people in the cabinet who back a second referendum, but they are riding several horses so they don’t have to quit.”

There is even talk among senior ministers that May should form a temporary government of national unity should she be defeated, as a “last throw of the dice” to find a majority for a Brexit plan that works.

Amber Rudd, the work and pensions secretary, broke ranks on Saturday to become the first cabinet minister to openly suggest alternatives to May’s plan. Rudd argued that a Norway-style soft Brexit could be the way through the impasse. She also said in an interview with the Times that she did not rule out a second referendum and made it clear that if there were to be one, she would vote remain.

Writing on the Guardian website, however, the justice secretary, David Gauke, dismisses both the Norway option and a second referendum, which he says would come with “great risks”. Strongly backing May’s deal, Gauke says a second referendum “is by no means guaranteed to be a silver bullet. In fact it is more likely to entrench division and lead to at least a further year of damaging uncertainty.”

Downing Street said it remained firmly opposed to a second referendum, and insisted the prime minister was focused on winning the “meaningful vote”. In an interview with the Mail on Sunday, May warned Tory rebels that the country would be in “uncharted waters” if the deal was rejected with “a very real risk of no Brexit or leaving the European Union with no deal”. She also said voting down the deal would risk handing the keys of No 10 to Jeremy Corbyn.

But a senior Conservative backbencher, aware of the level of opposition to May’s deal in the party, said there could be at least 100 Tory MPs who would not support her. A close ally of the former foreign secretary Boris Johnson said a defeat on that scale would leave May no choice but to resign. “If it is under 50 she can probably go on and try to ask for concessions from Brussels, then put it back to a second vote. If it’s 50 to 100 it is more difficult but if it is over 100 it is impossible to see how she can carry on. People are already saying it is like the last days of Rome and it cannot go on like this.”

In a further blow to the prime minister, the all-party select committee on exiting the European Union, which contains 10 Tory MPs, published a unanimous and scathing report on her deal on Sunday. The committee says many of the most important questions about the UK’s future relations with the EU have been left unanswered because of ministers’ unwillingness to confront key issues. The committee chairman, the Labour MP Hilary Benn, said the deal lacks clarity and would represent “a huge step into the unknown”.

The committee unites behind a conclusion that “there are no realistic, long-term proposals from the government to reconcile maintaining an open border on the island of Ireland with leaving the single market and customs union”. As far negotiations on an UK-EU trade deal is concerned it says negotiations could be further delayed and complicated “because the government has yet to set out clear objectives for the future relationship that are realistic, workable and have the support of parliament”.

On Sunday, the Tory MP Sarah Wollaston reveals a new strategy to secure cross-party backing for a second referendum. Rather than tabling an amendment to Tuesday’s main motion, as previously planned, she will launch a campaign to rally MPs behind a second vote if and when May’s deal is defeated. Tory MPs in favour of a second vote hope Labour will officially back the call allowing a parliamentary majority to be formed. Wollaston said: “If the vote in parliament goes ahead then, once rejected, I will be bringing a cross-party amendment at the earliest opportunity to press for a people’s vote. I urge the Labour frontbench to then stick to their promise to support it. No responsible government or opposition frontbench could knowingly unleash the dire consequences of crashing out with no deal and no transition and MPs on both sides must be realistic and honest with the public that no further concessions or alternative deals are on offer.”

Downing Street revealed that the environment secretary, Michael Gove, will close the debate for the government on Tuesday evening. Voting will take place at 7pm. On Tuesday morning, the Speaker, John Bercow, will select up to six amendments to the main government motion that seeks approval for May’s deal. One of these – tabled by Benn and with strong cross-party backing – calls both for the deal to be rejected and for parliament to rule out a no deal outcome. Senior MPs believe that if the Benn amendment is chosen by the speaker and voted on before the main motion, it could be passed, meaning the government motion would be superseded. The effect would be a double defeat for May, and narrow her options down, as she would not be able to proceed towards a no deal. Some senior MPs say this will make a second referendum more likely.

The Foreign Office minister Sir Alan Duncan said on Saturday night that May had to stay even if she was heavily defeated. “Whatever happens Theresa May must not resign,” he said. “The country needs her. No one can do any better. We need her to see it through. Nothing could be worse than a leadership contest while Brexit remains unresolved.”

Frans Timmermans, the vice-president of the European commission, speaking at a conference in Portugal attended by Jeremy Corbyn, on Saturday urged the Labour leader to back a second referendum. He said: “The Brexit vote was the lowest point in my political life. We respect the vote. But since that vote, much has changed in the world. And the EU has changed. Looking at the UK and having listened to Jeremy Corbyn and his plans, I have a question: Can you achieve what you want more easily as a member of the EU family or on your own?”