The Greater Manchester mayor said on X: “I have today written to the Chair of Labour’s National Executive Committee seeking permission to enter the selection process for a candidate for the forthcoming Gorton and Denton by-election.”
It comes after former minister Andrew Gwynne formally stepped down on Friday on health grounds.
In his letter to the NEC, which he shared on social media, Mr Burnham said the decision had been “difficult” but insisted he would support the work of the government “not undermine it” and that he had “passed on this assurance to the Prime Minister”.
The timetable for the Gorton and Denton by-election is short.
The deadline for applicants to put their name forward is midnight on Sunday, with a longlist of candidates set to be drawn up by a selection panel on Monday.
Shortlisting interviews will be held on Tuesday before hustings and selection will take place next Saturday.
Mr Burnham lost two previous Labour leadership challenges – against Ed Miliband in 2010 and Jeremy Corbyn in 2015.
If elected, Mr Burnham would be required by law to stand down as mayor of Greater Manchester, which would then trigger a city-wide by-election.
The NEC can still block his bid for selection by refusing permission, and supporters of Sir Keir Starmer are reported to be mobilising to prevent him from becoming a candidate.
But several senior Labour figures have called for Mr Burnham to be allowed to stand, with the party’s deputy leader, Lucy Powell, saying the decision should be up to local party members.
London mayor Sir Sadiq Khan said: “I think if Andy Burnham wants to be a member of Parliament, Andy Burnham should be allowed to be a member of Parliament.”
Ed Miliband, the energy secretary, said Mr Burnham would be “a massive asset” in parliament, and hoped the Gorton and Denton party members would have “the option” of selecting him as a candidate.
