Burnham, the Greater Manchester mayor, won a decisive victory for the governing Labour Party in an election for a parliamentary seat in northwest England,and has signalled that he will use his seat to enter any contest to replace Starmer.
Several Labour MPs said the prime minister should consider stepping down.
But Starmer, who won a landslide election in 2024, said he was "not going to walk away", reeling off a list of actions during his two years in power: closer ties to the European Union, stabilising the economy and reducing waiting times for the health service.
"If there is a contest ... then yes, I will run, I will stand, and I've said repeatedly I'm not going to walk away," Starmer told reporters in London on Friday during a visit designed to indicate it was "business as usual" for him.
He again warned of the dangers of a potentially disruptive leadership campaign.
His resistance to growing Labour calls to set a timetable to step down could threaten hopes of an orderly transition by putting divisions on public display in a leadership contest.
Burnham won the contest in Makerfield in northwest England with 54.8 per cent of the vote, beating the candidate for the populist Reform UK, on 34.5 per cent, and boosting his image as someone who could halt the rise of veteran Brexit campaigner Nigel Farage's party.
His victory not only sent him back to parliament, but also boosted the hopes of some worried Labour MPs that they can win the next national election, due in 2029, something some Labour MPs say Starmer, struggling with some of the worst popularity ratings of any British leader, cannot achieve.
Burnham used his victory speech in Makerfield to say he wanted to counter the rise of polarising, populist politics and turn Britain "away from the path that takes us to a divided, dark politics of the kind we see in the United States".
He hailed the result as a "turning point" for Labour.
"We must hear it, we must act upon it, and we must get it right," he said.
"There will be no second chance."
Burnham, a 56-year-old career politician, has backed the nationalisation of key public services and criticised what he called four decades of failed neo-liberal economics.
Polls indicate he would win a formal leadership contest.
That would mean Britain installing its seventh prime minister in about a decade, the highest turnover in nearly two centuries - a reflection of voter anger at successive failures to improve living standards and public services and tackle illegal immigration.
Starmer, 63, has repeatedly vowed to fight on despite scandals, policy U-turns and accusations of indecision, wanting to finish his five-year term by fulfilling his promise to solve some of Britain's most pressing problems.
But about a quarter of his MPs have urged him to quit since Labour suffered heavy losses in local elections in May, and senior colleagues, including the defence and health ministers, have resigned.
Several Labour MPs said the scale of Burnham's win should force Starmer to consider stepping aside.
Many Labour MPs fear losing their seats in the next election to Farage's party, which leads opinion polls.
Another of Starmer's rivals, former health minister Wes Streeting, said he would force a contest soon unless the prime minister announced when he would stand down.
He said Burnham's victory was proof that Labour needed to change.
