Speaking in a BBC debate with other contenders in an election to a parliamentary seat in northern England, Greater Manchester mayor Burnham said the governing Labour Party, which is trailing in opinion polls, needed a fundamental change.
Starmer, whose popularity ratings are some of the lowest for a British leader, has vowed to fight to keep his job, but potential rivals are circling him, including Burnham and former health minister Wes Streeting, who resigned last month in protest at the prime minister's record.
Asked about his ambitions, Burnham, who lost two earlier attempts to become Labour leader, said he did not want to get "ahead of himself", acknowledging he must first win the election in Makerfield on June 18 to be able to challenge Starmer.
"I can't do anything unless I'm lucky enough to get the support of people here. But if I get your support, I would seek to represent you at the highest possible level," he said in the debate, according to a BBC transcript.
Burnham said he thought Streeting seemed to have already "launched a leadership contest", something the former health minister has not yet done formally.
"So if that is running, I would seek to join it. But I'd have to persuade members of the parliamentary Labour Party to do the same. So that's the only question... I've said to my team, let's have a proper look at this and let's develop a policy," he said.
Hitting back at Burnham, Downing Street said the process for challenging the Labour leader "has not been triggered", and Starmer "will not walk away".
A No 10 spokesperson said: "With Keir Starmer's leadership, this Labour government is supporting people with the cost of living, cutting NHS waiting lists, restoring control of our borders and lifting half a million children out of poverty.
"The country expects us to focus on governing and to deliver change for hard-working people, not get distracted by Westminster debates.
"The Labour Party has a process for challenging a leader and it has not been triggered.
"The prime minister will not walk away from the mandate he was given just two years ago to build a stronger, fairer Britain."
with PA
