Andy Burnham will arrive in Westminster on Monday ahead of crunch talks with Keir Starmer about his leadership.
The Makerfield MP will be sworn in as an MP next week, and plans to speak with the PM to ask him to set out a timetable for his departure.
Mr Starmer has repeatedly vowed to fight any leadership challenge, insisting he will not “ walk away ”.
But the scale of Mr Burnham’s by-election victory in Makerfield has prompted more backbenchers and Labour grandees to call for the PM to stand down.
OPINION
View 2 ImagesKeir Starmer and Andy Burnham are expected to speak next week(Image: Sean Hansford | Manchester Evening News)
This newspaper spoke to several MPs who had signed a statement rejecting calls for a leadership election last month and have now changed their minds, while former home secretary Alan Johnson told LBC Mr Starmer must step aside.
He said: “I’d say it’s over, Keir. And listen, he’s a bright guy. He knows it’s over. He will forever be in the history books as the man who turned us around, Labour, from the second-worst result in our history to the second best result in our history in one five-year period.”
Those trying to oust the PM claim it is in the interests of the party for him to set out a timetable for his departure, arguing an extended contest would damage the Government.
A Labour source said: “Both Andy and Wes are giving the PM the space to come to the right conclusion. Things can go in two directions now. One is orderly and comradely transition, the other is nasty, divisive and not good for the party or candidates.
“The question of what happens next is secondary to the PM deciding to step down.
Things have changed, unlike before the Makerfield by-election, both change options are now preferable to the Parliamentary Labour Party over the status quo.”
Another Labour source said: “The people egging Keir to fight don’t have his best interests at heart, they’re factional fighters. He protects his legacy by doing the right thing for the country and party.”
It came as broadcaster and Labour peer Ayesha Hazarika claimed the PM would be remembered kindly if he stood down.
Writing in this newspaper, she said: “The critical question is who can best beat Farage at the ballot box And with genuine regret, that person is not the current Prime Minister. He achieved the historic feat of getting Labour into power for which so many are grateful. But no one man is bigger than the movement.
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“If he agrees to step aside, he could craft his political legacy with dignity. As the guy who refreshingly didn’t let his personal ambition and anger cloud his judgement about what was fundamentally right. Ironically, he would probably be instantly lionised.”
The PM and his backers have also stressed the need to focus on the by-election for the Greater Manchester mayoralty triggered by Mr Burnham’s election.
That by-election is scheduled to take place on July 30, with Labour set to announce its candidate on June 26.