Struggling with some of the lowest popularity ratings for any United Kingdom prime minister in modern political history, Starmer could decide as soon as Monday whether to step aside or fight a leadership contest against Burnham, one source said.
The scale of victory Burnham won for a parliamentary seat in northwestern England on Friday has piled pressure on Starmer, with dozens of MPs and some ministers privately calling for him to set out a timetable for his departure to clear the way for the former Greater Manchester mayor.
A source with knowledge of the matter said Starmer was spending the weekend thinking about and discussing his position with his family but that an expected conversation with Burnham would clarify matters.
"Keir likes to think about things," the source said.
Adding to the pressure on Starmer, US President Donald Trump predicted on his Truth Social platform that "Keir Starmer will resign as Prime Minister of the United Kingdom".
Trump then repeated his view that Starmer had "failed badly" on cutting immigration and boosting North Sea oil output.
Starmer's unpopularity was laid bare by Labour's heavy losses in local elections in May, and polls of party members indicate Burnham would win a formal leadership contest.
Should Burnham take the helm, he would become the UK's seventh prime minister in the past 10 years.
Sky News reported that it understood foreign minister Yvette Cooper had called on Starmer to stand down in a private conservation over the weekend.
Her spokesperson did not immediately respond to a request for comment.
Her apparent appeal, alongside other ministers and dozens of MPs, increased the sense that it is now a case of when rather than if Starmer would step aside.
Starmer said only a few days ago that he would stand in any formal Labour leadership contest that sought to replace him.
While Starmer's team believes his landslide general election win in 2024 gives him the mandate to stay in post until 2029, business minister Peter Kyle said the prime minister was reflecting on the "the political challenges that he faces in this moment".
Kyle said he had spoken to Starmer on Friday and had found a man who was questioning what "the country expected of him".
The conversation showed Starmer was in "very difficult circumstances," the business minister said.
"So I'm not going to deny the political challenges that he faces in this moment but what I'm also not going to do is say there is ever anything inevitable about the days ahead," Kyle told LBC radio.
"I don't know what's going to unfold in the days that lie ahead," Kyle told the BBC.
"I don't know the decisions that he will make on behalf of our country as he considers these issues, and then makes decisions.
"But we will find a way for this to be whatever unfolds, a functional process, one where the Labour Party is seen to put the interests of the country first and foremost, and we will carry on delivering for this country through whatever does unfold in the days ahead," Kyle said.
Burnham's thumping win over the populist Reform UK party to take a parliamentary seat in Makerfield prompted more MPs and ministers to press the prime minister to set a timetable for his departure to avoid what could be a divisive leadership race.
The team supporting Burnham, a 56-year-old career politician, had said they were giving Starmer the weekend to consider his position in the hope that he would set out an orderly transfer of power.
As yet, there was no indication the two had spoken.
Former minister Jess Phillips - who is a supporter of health secretary Wes Streeting, another potential challenger to Starmer - told the BBC that "it feels like we've come to the end of the road" and that it would be best for Starmer's departure to be "as dignified as possible".
The prime minister's office said on Sunday Starmer's position remained unchanged from Friday, when he said he will not "walk away" from Downing Street and will stand in any potential leadership contest.
with PA
