After winning the Makerfield by-election, Burnham will be sworn in on Monday as the Greater Manchester constituency's MP and is expected to press Starmer to set out plans to stand down.
Starmer spent the weekend at Chequers, the prime minister's country estate, contemplating his political future following mounting pressure from his Labour Party colleagues.
He is reported to have been told by members of his Cabinet, including Foreign Secretary Yvette Cooper, that he should set out a timetable for his departure.
No 10 said Starmer's position remained unchanged from Friday, when he said he would not "walk away" from Downing Street and would stand in any potential contest.
But Cabinet minister Peter Kyle did not shut down questions of Starmer considering quitting when he spoke to broadcasters on Sunday.
The Business Secretary said Starmer was taking time to "think through what the political realities are today compared to last week, the week before".
"I don't know the decisions that he will make on behalf of our country", but that "whatever unfolds" in the coming days will be a "functional process", he said.
It is understood Burnham is looking for a transition of about a month, which would give Starmer time to get his Defence Investment Plan over the line in time for the upcoming NATO summit and sign his planned EU co-operation deal at the second annual reset summit with the bloc in July.
It is thought Burnham's camp would however accept a timetable that sees Starmer in No 10 until September.
Questions are being raised within Labour about how a change of leader would come about - with some backing a leadership contest and others a swift transfer of power to Burnham.
The number of Labour MPs calling for Starmer to go has topped 100 - just under a quarter of the party's cohort in the Commons - and includes some who signed a letter warning against a leadership contest just last month.
with PA
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