The British prime minister has repeatedly vowed to fight any leadership challenge, insisting he will not "walk away".
But Burnham's by-election victory has prompted more backbenchers and Labour grandees to call for Starmer to stand down.
Some MPs who had signed a statement rejecting calls for a leadership election in May have now reversed their position while former home secretary Alan Johnson told LBC his message to the prime minister would be: "It's over, Keir."
Starmer is understood to have spoken to a number of cabinet ministers on Friday, some of whom are reported to have told him he should set out a timetable for his departure.
Some in Westminster believe a contest could begin as early as next week but allies of Burnham favour a longer wait to allow them to prepare for government.
Former transport secretary Louise Haigh, one of his supporters, said after his by-election victory: "We really hope that this can be a managed and orderly transition and Keir Starmer will reflect on the results, and Andy and Keir can meet in the coming days, and over the next week, and agree a path forward."
It is understood that Burnham's camp wants Starmer to set out his plans in the coming days but would accept a timetable that kept him in No 10 until September.
But former deputy leader Baroness Harriet Harman has urged the party to move faster, telling Sky News's Electoral Dysfunction podcast ministers could not be left "in a state of paralysis all through the summer".
Meanwhile, former home secretary Alan Johnson told LBC that Starmer now has to 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."
But Starmer appeared to be digging in, warning Labour staffers during a call on Friday lunchtime to avoid "plunging our party and our country into chaos by turning on each other and tearing apart our party and our movement".
It is understood he has also amassed a campaign war chest to fight any leadership challenge with the backing of a group of private donors, as first reported by The Times.
Fundraising has ramped up in the last two days with total pledges running into six figures, sources said.
Starmer and his backers in the Parliamentary Labour Party have also stressed the need to focus on the by-election for the Greater Manchester mayoralty triggered by Burnham's election.
That by-election is scheduled to take place on July 30, with Labour set to announce its candidate next Friday.
The party is already pitching the contest as a two-horse race between Labour and Reform UK but the Greens will look to build on their success in the Gorton and Denton by-election with a strong challenge of their own.
Green leader Zack Polanski has sought to portray his party as the main challenger to Nigel Farage, saying the mayoral contest will be "a straight up battle between the Green Party vs Reform".
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