Senior Tories have given Theresa May just three months to improve or face a renewed attempt to oust her, amid concerns she is leading the party towards “utter destruction”.

In a sign of the fragility of the prime minister’s leadership after another week of cabinet infighting, figures inside the party said that a disastrous performance in May’s local elections could trigger a no-confidence vote.

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There are concerns they will produce a Tory meltdown in London. “There’s definitely been a further shift against her,” said one ex-minister. “If people could wave a magic wand tomorrow, she would be gone ... if we get wiped out in London, then more will say things cannot get any better under her. The alternatives would be the utter destruction of the Tory party or a chaotic leadership election, which would at least offer some way out.”

A former cabinet minister said: “It looks like these elections are going to be very bad. We could well be wiped out in London and cities like Birmingham because there is no reason to vote Tory. MPs think in two ways: what can I do to save my seat, and what’s in it for me? I think something will then happen. People will begin to manoeuvre over the leadership. There will be a degree of panic.”

Both foreign secretary Boris Johnson and defence secretary Gavin Williamson face criticism from colleagues over claims they are jostling for position. There has been a backlash against Williamson from fellow frontbenchers after he issued a pointed attack against Russia on the same day as revealing a past infidelity with a co-worker while a manager at fireplace maker Elgin and Hall.

Some defence figures are understood to have been annoyed by his claim that Russia could cause “thousands and thousands and thousands” of deaths by damaging British infrastructure. May appointed Williamson defence secretary in November. Some ministers believe May should now shore up her position by reiterating her post-election pledge to step aside willingly when MPs believe they are ready for a new leader. Meanwhile, party veterans are trying to convince colleagues to “get Brexit out of the way” before she is challenged – a sentiment still popular among many mainstream MPs.

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A formal vote of no confidence in May would be held if Sir Graham Brady, the chairman of the backbench 1922 committee, receives letters from 48 MPs demanding a contest. He alone knows how many letters have been submitted, but Tories said this week that more had been despatched in the wake of this month’s chaotic reshuffle, which left many MPs confused and frustrated.

“The reshuffle was a total disaster and upset a lot of people – people fired, people moved sideways and people ignored,” said one former minister. “There is no way she will fight another election. Certainly more letters have gone in.”

The crisis has increased as May is now feeling the heat from new wings of the party. Leading pro-Brexit MPs had regarded the prime minister as the most likely to deliver the hard Brexit they envisage. However, some are reaching the view that May can no longer be relied upon and are already plotting a rebellion against the government later this year, when the EU withdrawal bill returns to the Commons.

“She’s caught between a rock and a hard place,” said another former minister. “The Brexiters would knife her tomorrow if she steps one iota out of their imagined version of Brexit, while the Remainers hold no loyalty towards her because of the way she treated them.”

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Amber Rudd, the home secretary, is understood to have led the reprimand of Johnson during this week’s cabinet after briefings that he would use the meeting to argue for a “Brexit dividend” for the NHS.

One party veteran said Johnson was still thinking about his chances. “The smart money may not be on him but he is a huge beast in the jungle and he cannot be ignored,” they said. “Some people think he is a liability but on the other hand he is very popular among sections of the membership, particularly women. If he is not handled properly he will go off the reservation.”

The environment secretary Michael Gove’s supporters were also limbering up for another push, he added. “There is some support for Govey.”