Keir Starmer faces a fight for his political life in the next 24 hours as potential Labour leadership rivals, from Wes Streeting to Angela Rayner, begin positioning themselves for a contest.

Starmer is hoping to save his job on Monday with a speech promising to “face up to the big challenges” for the country on growth, energy, defence and Europe.

However, his prospects of staying in No 10 appeared to be fading on Sunday after about 40 Labour MPs called for him to set a date to step down, including many backers of the Greater Manchester mayor, Andy Burnham, who hope to pressure the prime minister into resigning.

Leadership contenders were circling after a disastrous set of local election results in which the party lost support to Reform UK and the Greens. One ally of Streeting said: “Wes isn’t going to challenge Keir but he is preparing in case it all falls apart.”

The health secretary is understood to have delivered this same message to No 10 but he does not want be the first to make a move against the prime minister – even though some of his allies are pushing for a challenge after Starmer’s speech, believing it to be his best chance.

Rayner, the former deputy prime minister, set out her prescription for change and warned Starmer he needed to “meet the moment”. Her supporters say she is not set on being a candidate but is prepared for the possibility of a leadership run.

She also backed a return to parliament for Burnham, who would be the leading candidate to replace Starmer if he were allowed to run for a seat. If Burnham cannot return, there could be a race among potential candidates on the left to secure his endorsement.

Speaking on behalf of the government on Monday, Peter Kyle, the business secretary, said that while it was “possible” for Starmer to win back areas lost to Reform UK last week, he needed to show boldness in his speech.

“I do believe it is possible. I am not saying that it is easy, and I’m not saying it will happen. I’m saying it can,” he told Sky News. “And this is a moment where Keir will start to show … that he is willing to meet the challenge and the scale of challenge. But I accept also the window of opportunity has narrowed as a result of the elections last week … Therefore, he has to set out a direction and a scale of boldness in the speech.”

The government, Kyle said, had “dwelt too much in the challenge” when it entered office and created some pessimism, which needed to be countered, for example with new efforts over Europe.

Many of those calling on Starmer to set a date for his departure were supporters of Burnham. However, their demands appeared at risk of backfiring, as the momentum for a swift challenge to Starmer’s premiership would favour Streeting, Rayner or other cabinet contenders such as Ed Miliband.

Amid a febrile mood in the party, the Labour backbencher Catherine West pressed on with her plan to gather names for a “stalking horse” challenge if the prime minister does not set a timetable to quit – designed to persuade other candidates to come forward.

West is not widely considered a viable candidate herself but is seeking 80 supporters among Labour MPs to trigger an immediate contest. Despite wanting Starmer to go, Burnham’s backers have been trying to persuade her to withdraw her challenge as it would not leave him enough time to enter parliament. “It’s not gone to plan. This wasn’t meant to benefit Wes,” said one MP supporter of Burnham. Another MP described West’s intervention as chaotic and “like one of those free-running horses at the Grand National”.

Kyle said he did not support Burnham being allowed to return to parliament, saying Burnham had promised to see out his term as mayor. “I think these sorts of promises are really important,” he said.

Starmer will attempt to stop the speculation about his premiership with a speech on Monday promising to define his government by “putting Britain at the heart of Europe”.

“To meet the challenges that our country faces, incremental change won’t cut it,” he will say. “On growth, defence, Europe, energy – we need a bigger response than we anticipated in 2024, because these are not ordinary times.”

He will add: “This Labour government will be defined by rebuilding our relationship and by putting Britain at the heart of Europe. So that we are stronger on the economy, on trade, on defence, you name it.”

Bridget Phillipson, the education secretary, defended the prime minister on Sunday, telling the BBC it would be wrong to remove him, even though voters had given the party a “real kicking” at the ballot box and people felt “bitterly let down”.

Few Labour MPs appear to think Starmer can regain his authority. One Labour cabinet source said: “There is a residual loyalty to Keir but [the cabinet] are at end of their tether.”

Labour MPs are furious with the prime minister for his lacklustre response to the crisis engulfing Labour from its right and left flanks.

Facing a threat to his job, Starmer gave an interview to the Observer saying he wanted to serve for two terms or 10 years. He has also attempted to refresh his government by bringing back the former prime minister Gordon Brown as an adviser on finance, and the former Labour deputy leader Harriet Harman as an adviser on women and girls.

“He’s got the hide of a rhino, and he’s not reading the room. It’s not possible to see how he can recover from this,” said one MP who would like to see Burnham return.

MPs said any contest could be unpredictable, with the possibility that other ministers such as Phillipson, Yvette Cooper, Shabana Mahmood or Al Carns could take the opportunity to run.

In a sign MPs are readying for a big debate about the government’s future direction, the Labour Growth Group is this week planning to present its blueprint for “a new economic settlement” to No 10 and the party more widely, calling for higher capital gains tax to fund a 2p cut to national insurance, among other policies.

A government minister said: “Labour can still win in 2029, but only if we turn warm words about working people into radical and urgent action. Voters are clear they want to know whose side we’re on and what we’re prepared to change. These ideas should be taken seriously because they speak directly to the people Labour was elected to serve.”

Rayner also released a set of policy proposals for economic renewal as she made her first intervention since the election results. She said: “What we are doing isn’t working, and it needs to change. This may be our last chance.” She said Starmer “must now meet the moment and set out the change our country needs”, and called for an acknowledgment that it was wrong to block Burnham’s attempt to return to parliament.

Suggesting how Labour needed to change, she said the party was in “danger of becoming a party of the well-off” and described the Peter Mandelson scandal as having showed a “toxic culture of cronyism”.

Some on the left have been urging Ed Miliband to be their candidate as an alternative, believing Rayner does not have enough support in the country and given that the HMRC inquiry into her tax affairs is not yet settled.