Cabinet ministers loyal to Keir Starmer have told him that he faces being forced out of office by his party if he does not set a timetable for his departure by the end of the weekend.

Andy Burnham, who won a compelling majority in the Makerfield byelection overnight, is expected to travel to London on Monday to meet MPs in the expectation of becoming prime minister within weeks. One cabinet minister – who has not previously told the prime minister to go – said his departure was now “inevitable”.

One MP said they believed there were about 200 Labour MPs prepared – if necessary – to sign Burnham’s nomination papers for a challenge, though his supporters are hoping for a coronation.

Starmer called members of the cabinet on Friday afternoon to set out his determination to fight on. The transport secretary, Heidi Alexander, is said by sources to be among those who expressed concerns in a call on Friday.

At least two, Ed Miliband and Shabana Mahmood, have previously suggested to the prime minister that he should set out a timetable for his departure.

Other cabinet ministers are expected to press the prime minister about whether fighting a leadership contest would be wise. Another cabinet source said: “Everyone thinks it is over and everyone wants it to be a dignified, orderly exit.”

Makerfield election chart

Several cabinet ministers expressed a desire to show loyalty – while adding the caveat that they knew the situation made it very difficult for the prime minister to continue.

But another said: “There comes a point where you ask, what is more important? Is it loyalty or delivering your agenda? There are big agendas at stake here and those who don’t see which way this is going are going to inevitably end up looking like they were the last ones in the bunker.”

Two Labour grandees – David Blunkett and Harriet Harman – have also said there should be a timetable for new leadership.

Senior Labour sources said they believed that if the prime minister did not resign over the weekend – or indicate that he would allow a transition to a new leader – then there would be an intervention at Tuesday’s cabinet meeting.

Starmer says he will stand if there is a leadership contest after Burnham’s win – video

“The prime minister cannot pull the same move again where he refuses to talk to his own cabinet about his future,” one senior source said. “He has a choice of allowing his cabinet and ministers to show open support for his rivals or risk the same situation as Boris Johnson, where you have three education secretaries in three days.”

But some of Starmer’s closest allies said the prime minister had the most agency, despite Burnham’s momentum, because the new MP was seeking to avoid the internally damaging move of mounting a leadership challenge.

“In the aftermath of the locals, it was Keir who actually held all the cards. Burnham’s people do not want a contest, they want Keir to do what Burnham wants. Well, that’s up to Keir if he decides he wants to do that and he actually does not have to do anything they want,” one said.

A pro-Starmer memo circulating among loyalist MPs shows the attack arguments the prime minister and his team would be likely to make in a leadership campaign.

Has Burnham’s win in Makerfield sealed Starmer’s fate? – The Latest

The memo, seen by the Guardian, says: “[Burnham] hasn’t faced any real scrutiny yet. A true contest would expose him to questions that he hasn’t ever before had to answer and likely see his support wane as a result.” It also argues that in polling terms “the trajectory for AB has not been positive” with his favourability dropping, and arguing “the membership can change their view”.

The existence of a memo drafted by allies of Starmer reveals his preparations for a contest are under way, but also underlines the risks of a wounding civil war within the party with each side trying to “expose” the other.

Starmer has told reporters at an event in north London: “If there is a contest, just to be clear with you, then, yes, I will run.”

He also warned that such a contest would “plunge us into chaos” and that Labour needed to “pull together” to contest the byelection to replace Burnham as the mayor of Greater Manchester.

Louise Haigh, the former transport secretary who helped lead Burnham’s campaign, called for Starmer to avoid what would be a “brutal and unpleasant” leadership contest and set out a timetable for his exit.

One MP close to the Burnham campaign said the prime minister needed to “get it out of his system” before he reached the “inevitable” conclusion.

Andy Burnham and supporters celebrating at Ashton Town FC the morning after winning the Makerfield byelection. Photograph: Christopher Thomond/The Guardian

A supporter of Wes Streeting – the former health secretary who has said it is also his intention to challenge Starmer – said it was likely that Streeting and Burnham would “thrash something out”, with the most plausible outcome that Streeting could agree to back Burnham in return for the promise of a major role.

The two men already spoke earlier in the week but have not yet had a crunch conversation, while still hoping that Starmer could resign of his own accord.

Streeting has told supporters that a leadership contest should still go ahead, but it can only be “comradely” if the prime minister does not take part.

Several backers of Streeting said they believed it was inevitable Burnham would become prime minister – given his 9,000 majority in Makerfield – but that he should still be tested in an accelerated contest that should conclude before the Greater Manchester mayoralty byelection.

Angela Rayner, the former deputy prime minister who had been widely expected to be a leadership contender, will not run in any contest where Burnham challenges the prime minister.

Starmer’s operation has a spreadsheet of MPs loyal to his leadership and they believe they still have enough support to match a challenge from Burnham.

One Starmer ally said Burnham’s victory rally showed he did not have “fresh ideas” and all of the policies he announced were already being implemented by the government, from apprenticeships to lower bus fares.

Makerfield election chart

“Obviously, winning 25,000 votes in a byelection is great, but what about the 10m votes we won in an election two years ago, which gave the PM the mandate to govern?” they said.

With a contest looming on the horizon if Starmer does not agree to hand power to Burnham, his team are currently looking at office space for a leadership campaign, are working on messaging and events in the early days, and have raised more than £100,000 in donations in preparation.

Another Starmer supporter in the government said: “He will fight it and he will win.”

Andy Burnham: Makerfield victory is ‘our last chance’ to change Britain – video

In his victory speech, Burnham said the result “could be a turning point” and that people had “voted for change. They have voted for more power for the north and everywhere forgotten by Westminster.”