Boris Johnson has been put on notice by rebel Conservatives that he will not survive long as prime minister unless he drops his no-deal Brexit agenda, as he stands on the brink of entering Downing Street.

Johnson is expected to be announced as the choice of the Conservative party to be the next prime minister on Tuesday morning after running a campaign against Jeremy Hunt that promised to take the UK out of the EU by the end of October “do or die”.

On the eve of his probable coronation by the Tory membership, several senior Conservatives sent warning shots that his hardline Brexit plans put him on a collision course with parliament.

Rory Stewart, a former leadership rival, joined Philip Hammond and David Gauke in telling Johnson he would quit the cabinet before the new prime minister takes office rather than serve under him.

In a further ominous move for Johnson, Sir Alan Duncan, a Foreign Office minister, resigned dramatically to launch an attempt to test whether the new prime minister could command a majority among MPs.

Duncan’s proposal for an emergency House of Commons debate on support for Johnson was turned down by the Speaker, but it was a sign that some Conservatives are already organising to make life difficult for the new incumbent of No 10.

Duncan said he had “very grave concerns that Johnson flies by the seat of his pants” and branded his former boss “haphazard and ramshackle”, but publicly insisted he was trying to be helpful to the new prime minister by ending speculation about whether MPs supported the new incumbent in No 10.

The motion stated: “That this House has considered the merits of the newly chosen leader of the Conservative party, and supports his wish to form a government.”

However, it opened up the possibility that Johnson might fail to demonstrate he had the support of parliament, leaving Theresa May potentially unable to recommend him as her successor to the Queen on Wednesday.

Conservative backbenchers fighting no deal are not planning to bring down Johnson immediately, with one serious critic of the frontrunner saying the party “would take a very, very dim view of getting rid of him before he has done anything yet”.

But several Conservative MPs said they would regard Johnson’s first speech to the nation and cabinet appointments as a test of whether he was capable of reaching out across parliament to find a majority for a plan to leave the EU that can find approval from Eurosceptics and more moderate Brexit supporters.

If he does not, then organisation will start again in earnest to prevent him pursuing a no-deal Brexit, with some senior Tories already sending out feelers about the possibility of a “national unity” government with opposition MPs.

A senior source on Johnson’s transition team said that, as the new Tory leader, he would immediately emphasise “unity” and the need to bring the party and the country back together.

The rebels are prepared to give Johnson the summer to see if he can make headway towards coming to a fresh agreement with the EU that avoids no deal but will make their moves against him after that if he takes the government full tilt towards a no-deal Brexit.

One former minister said a “sizeable chunk” of the 42 Tories who voted against a no-deal Brexit last week were prepared to put their own careers on the line to stop Johnson pursuing that path – either through a legislative block on leaving the EU without a deal or a confidence vote if that proves impossible.

He said what Johnson said on the steps of Downing Street on Wednesday would be crucial, as his critics within the party will be infuriated and galvanised to action if he “tries to act like he has a majority of 150” or goes down the path of “Trumpian rhetoric” that forgets the country is split on Brexit.

Another rebel Tory, Philip Lee, a former health minister pushing for a second referendum, said: “It is going to be really challenging for him to govern. I don’t see how he has got a majority to govern. It’s a very, very fragile situation with only a tiny majority.”

Johnson will have a majority of just two if the party loses the Brecon and Radnorshire byelection this month to the Liberal Democrats. His majority was eroded further on Monday by the withdrawal of the whip from Charlie Elphicke, a Tory backbencher charged with three counts of sexual assault against two women.

Labour have backed away from submitting a confidence motion in Johnson this week as they believe the Tory rebels are prepared to give him a chance, but there is a likelihood they will lay one in the weeks after parliament returns from recess. Then Conservative MPs against no deal will have to make a choice about whether to vote with the opposition against Johnson and risk triggering a general election that could put Jeremy Corbyn into No 10.

If Johnson were to lose a confidence motion, some Conservative MPs are openly talking about the possibility of doing a deal with some Labour MPs to form a national unity government led by a respected figure such as the shadow Brexit secretary, Keir Starmer, in order to secure a second referendum pitting remain against a deal.

Nick Boles, a former Tory MP who resigned to be an independent, appeared to flirt with more cross-party alliances as he praised Starmer for saying: “We are in the middle of the most intense political crisis for a generation; Boris Johnson as PM will make things worse not better. The national interest lies in MPs from all parties working together to stop him.”

Johnson is expected to make his choices for the top four cabinet jobs on Tuesday night, the day before May goes to Buckingham Palace to formally resign to the Queen.

Another former Tory minister said moderates in the party would be watching his cabinet appointments closely amid deep concerns about Johnson appointing hardliners such as Priti Patel, Liz Truss or Jacob Rees-Mogg to very senior roles in cabinet.

While Tory moderates are biding their time, Eurosceptics are also circling Johnson to ensure he keeps to his promise of taking the UK out of the EU by 31 October and ditching May’s Brexit deal. One senior Eurosceptic on Johnson’s campaign team said they were perfectly prepared to “take him out ourselves” if he failed on his promise to deliver a clear-cut Brexit by that date.

May held her farewell drinks in Downing Street on Monday night with MPs from across the spectrum of the Tory party and the Democratic Unionist party in attendance, telling them in a short speech that their priority was stopping Corbyn entering Downing Street.