As a directly-elected mayor, Burnham has to get approval from Labour’s ruling national executive committee before he can enter the race to be the party’s candidate. That decision is expected on Sunday.
Several Labour MPs have reacted angrily to the suggestion that Burnham could be blocked from standing in the by-election.
Burnham had until Saturday afternoon to submit his application and on the dot of the 5pm deadline, he shared his letter, external to the NEC on social media, ending days of speculation over what he was planning.
In his letter, Burnham said there was a “direct threat to everything Greater Manchester has always been about from a brand of politics which seeks to pit people against each other”.
“I see this by-election as the frontline of that fight for the Manchester Way and I feel I owe it to a city which has given me so much to lead it from the front, despite the risks involved.”
He said he had left Westminster nearly a decade ago because he believed “it wasn’t working for people in our part of the world” and that as mayor he had “tried to pioneer a different way of doing things”.
However, he added that he believed Manchester “won’t be able to be everything it should be without similar changes at a national level.
“This is why I feel the need to go back.”
There has been speculation that the prime minister could face a leadership challenge, after the May elections, which will see votes take place in the Welsh Parliament, the Scottish Parliament and in some local councils in England.
Burnham has been touted as a possible leadership contender but can only mount a challenge if he is an MP in the Commons.
In his letter to the NEC, Burnham insisted he would be in Parliament “to support the work of the government, not undermine it”.
“I have passed this assurance on to the prime minister.”
