Starmer has publicly vowed to stay in post, but pressure is building as more and more Labour Party colleagues conclude his time is up.
Expectation is growing he will announce a timetable for his resignation as soon as Monday.
That's the day Burnham will be sworn in as an MP in the House of Commons after winning a by-election last week.
Business Secretary Peter Kyle said on Sunday that Starmer was "making time to reflect on the political realities, challenges and opportunities that he finds himself in".
"I know he is a prime minister who always puts his country first," Kyle told the BBC, though he said reports Starmer would resign were "speculation".
Starmer is spending the weekend at Chequers, the country mansion used by British prime ministers, with his family. He gave no public hint about his decision, but sent a Father's Day message on social media.
"Being a dad is my greatest joy. Today, I'm thinking about my dad, and the father I am to my children because of him," he wrote on X.
If Starmer quits, he will be the sixth prime minister to leave office in the past 10 years, an extraordinary rate of churn for the United Kingdom.
Discontent with the prime minister has been building for months, with Labour MPs desperate to reverse the government's decline in popularity since Starmer led the centre-left party to a landslide election victory in July 2024.
He has struggled to deliver promised economic growth, repair tattered public services and ease the cost of living, and has been hamstrung by repeated missteps, including his decision to appoint Peter Mandelson, a scandal-tarnished friend of Jeffrey Epstein, as the UK ambassador to the United States.
Labour is losing liberal voters to the growing Green Party and facing a rising Reform UK, the Nigel Farage-led anti-immigration party that consistently leads in nationwide opinion polls.
Burnham, until this week the popular mayor of Greater Manchester, decisively won the seat of Makerfield in northwestern England in a by-election held on Thursday. He took almost 55 per cent of the 45,510 votes cast, over 9000 more than the Reform UK runner-up.
Now that he is an MP, he's in a position to challenge Starmer for leadership of the Labour Party. Burnham's acceptance speech left no doubt that he wants to lead both the party and the nation.
"Everyone knows that politics isn't working," he said.
"Everyone can feel that the country isn't where it should be. Tonight could, just could, be the turning point."
It's unclear whether Burnham would face a coronation or a challenge if Starmer steps aside.
Wes Streeting, who resigned as health secretary in May to protest against Starmer's leadership, has said he will run in a contest if there is one.
Starmer congratulated Burnham on Friday, but insisted he would fight any attempt to oust him.
"I will run, I will stand," if there is a Labour leadership contest, Starmer said. "I've said repeatedly I'm not going to walk away from that."
7°C