Burnham, a former Labour minister, has not confirmed whether he will apply to stand in the seat.

Asked on Thursday about the vacancy in the Greater Manchester constituency, he said he was “focused” on his job as mayor and cautioned people not to “rush to conclusions”.

Deputy Labour leader Lucy Powell, who sits on the NEC, appeared to indicate her support for Burnham telling an audience in London that she wanted “the very best candidate, the person who can most help Labour win in that by-election” selected.

She described Burnham as “an incredibly popular politician” and said he would have to “make up his own mind” about whether to stand.

Powell, who represents a Manchester constituency, said Labour members should “get behind Keir Starmer” adding that she didn’t want to see the by-election emerge into “in-fighting and talking about ourselves”.

There has been speculation that Sir Keir could face a leadership challenge after elections in May for the Welsh Parliament, Scottish Parliament and many local councils in England, which are expected to be difficult for Labour.

Nearly 30 councils have opted to delay elections this year due to a major reorganisation of local government.

Opposition MPs have criticised the delay, saying it would potentially disenfranchise millions of voters.

In order to enter any leadership race, Burnham – who has run twice before to be Labour leader – would have to be an MP.

He is expected to receive support from former deputy prime minister Angela Rayner, who the BBC understands will tell a Labour north west regional conference that the NEC should not block Burnham’s application.

Although Rayner resigned from government last year she still has strong support among some backbench MPs and members.

Several Labour sources have said they believe the NEC will block Burnham’s application to stand on the grounds that it would trigger an election for a new Manchester mayor, which would be expensive both for the party and the taxpayer.

The last mayoral election cost the Greater Manchester Combined Authority around £4.7m.