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.
He has insisted he would be in Parliament “to support the work of the government” and says he has assured the prime minister of this.
Labour MPs are split on the issue, some think Burnham should be allowed to stand as a strong candidate but others think having to contest his mayoralty opens the door to Reform potentially winning that.
London Mayor Sir Sadiq Khan said he should be allowed to be an MP if he wanted to be, while deputy Labour leader Lucy Powell said the by-election would be “difficult” and she wanted “to make sure we are putting the best team out on the pitch week after week”.
She called for Labour members to “get behind Keir Starmer”, adding that she did not want to see the by-election develop into “in-fighting and talking about ourselves”.
Energy Secretary Ed Miliband said Burnham would be “a massive asset” in Parliament and that he hoped Gorton and Denton party members would have “the option” of selecting him as a candidate.
But Health Secretary Wes Streeting was more cautious, saying there would be big questions to weigh up including what would happen if there was a mayoral election in Greater Manchester.
If Burnham were to be selected and won the Gorton and Denton seat, he would have to resign as Greater Manchester mayor, triggering a new election for that role.
Several Labour sources have said the NEC could block Burnham’s application on the grounds that a mayoral election would be expensive for the party.
It would also be costly for the taxpayer with the last mayoral election costing around £4.7m.
The area of Gorton and Denton has traditionally voted Labour and the party won the seat in 2024 with a majority of 13,000.
However, since then the party’s popularity has declined and Labour could face challenges from Reform UK and the Green Party who came second and third at the general election.
John McTernan, former political secretary to Tony Blair, said he thought Burnham was putting himself forward in the interests of the party.
He told BBC Breakfast: “Support for the party has nearly halved since the last general election. Andy is very popular in Greater Manchester and if Reform formed a government eventually it would be a challenge not just for Labour but for everything he and other mayors have achieved around England.”
Labour MP Jo White told the same programme Burnham may be the best candidate for the job but he had to consider his legacy as Greater Manchester mayor and if he became an MP he should work to ensure Labour retained the mayoralty.
“Is he willing to sacrifice giving up the Manchester mayor to Reform or another party to pursue his parliamentary ambitions?” she said.