Burnham, who is one of the governing Labour Party's most prominent figures and is currently mayor of Greater Manchester in northwest England, said he had put his name forward to contest a vacant parliamentary seat.
"This has been a difficult decision for me to make," he said in a letter to the party's National Executive Committee, adding that, if elected, he would "support the work of the government, not undermine it".
"I have passed on this assurance to the prime minister," he said.
Starmer's party is faring badly in opinion polls and faces a test in local elections in May, leading to speculation about a possible leadership challenge.
Polls also suggest Starmer himself is hugely unpopular, with a net favourability rating of minus 54, according to YouGov in December.
The same survey showed Burham was one of the few politicians without a negative net favourability rating.
Burnham, who was briefly the United Kingdom's deputy finance minister under Gordon Brown's Labour government in the 2000s, is hoping Labour will now choose him to contest the seat of Gorton and Denton after the current incumbent, former minister Andrew Gwynne, said this week he would step down.
He will not automatically be selected as Labour's candidate for the seat, which the party won comfortably in the 2024 general election, with media reports suggesting allies of Starmer might seek to block him.
Even if Burnham is selected to contest the Gorton and Denton seat he could find it hard to win, with one recent poll putting the populist Reform UK party of Nigel Farage ahead of Labour in the constituency.
