Burnham does not have to decide straight away if he wants to throw his hat into the ring, as Gwynne has yet to formally resign his seat but he is likely to face mounting questions about whether he intends to go for it or not.
If he does decide to seek selection he is likely to face competition from other Labour figues and former MPs who have their eye on the seat.
As a sitting mayor, he would also face the added complication of needing the approval of the NEC, under Labour Party rules.
Several Labour sources said they expected the committee to object on the grounds that Burnham standing for Parliament would in turn trigger an election for the Greater Manchester mayoralty, which would be seen as a drain on party funds as well as politically risky.
It is also possible that the NEC would insist that there should be an all-women shortlist of possible Labour candidates for a by-election because the majority of the party’s MPs are currently men.
But if the NEC blocked Burnham from standing, his allies among Labour MPs could force party officials to overturn that decision.
Health Minister Stephen Kinnock told BBC Breakfast that Burnham was “an incredibly talented and effective leader as the Mayor of Greater Manchester”.
He said picking a candidate “will be like any other selection process” and that the “rules and procedures will be set out in due course”.
Burnham has not yet commented on whether he will apply to be Labour’s candidate in the by-election in Gorton and Denton.
Speaking on Thursday, he said people “shouldn’t rush to conclusions”, and while he insisted he was “very focused on my role as mayor”, he did not rule out an attempt to return to Westminster.
Sir Keir has not commented on whether Burnham should stand, saying the NEC would set out the process for choosing a candidate, adding the mayor was “doing an excellent job” in Manchester.