Allies of Andy Burnham and Wes Streeting will ask cabinet ministers, friends and Labour grandees to persuade Keir Starmer over the weekend not to fight a leadership challenge.

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 MP said they believed there were about 200 Labour MPs prepared – if necessary – to sign Burnham’s nomination papers for a challenge.

Starmer was said to be calling members of the cabinet on Friday afternoon to set out his determination to fight on. 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.

One source said they believed some other cabinet ministers would press the prime minister about whether fighting a leadership contest would be wise.

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.

“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.”

Starmer has insisted he will not resign as prime minister and he would fight any leadership contest. The prime minister 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 reaches the “inevitable” conclusion.

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 as a leadership contender, will not run in any contest where Burnham challenges the prime minister.

Starmer is speaking to members of his cabinet and MPs over the weekend as he seeks to shore up support. His operation has a spreadsheet of MPs loyal to his leadership and they believe they still have enough support to match a challenge against Burnham.

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

“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.”

However, other Downing Street loyalists sounded less convinced the prime minister could stay, saying he would “fight on” and that Burnham was “not ready” but it was “not clear how it would all end”.

Rachel Reeves, Darren Jones, David Lammy and Steve Reed are among the cabinet ministers who are still backing his premiership, but others in the cabinet and more junior ministerial ranks have been quieter about what they think should happen next.

MPs are unsure how the next few days will play out, but one former minister said they thought Burnham and Streeting would try to thrash things out in talks, while also attempting to persuade Starmer to stand down.

They said other cabinet ministers would also “try and reason with the PM today and tomorrow, and if he doesn’t budge then a move against him happens Monday or Tuesday”. They said “almost everyone agrees … that Keir needs to do the right thing this weekend” by stepping down.

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.”

At a later rally, he in effect set out a programme for government and said “the word Makerfield in the future must be known as a byword for the change that came to British politics”.

“I think we need in this country right now for people to feel a sense of hope that there is something better to work towards on the horizon,” he said.