Deputy Prime Minister Grant Robertson ruled himself out, saying his position had not changed from when he lost his 2014 leadership challenge to Andrew Little.
He said his mind would not be changed.
Deputy Labour leader Kelvin Davis said he would not mount a challenge and instead would support whoever was selected by the caucus.
That leaves Chris Hipkins, Dr Megan Woods, Michael Wood and Kiri Allan as likely frontrunners.
Mr Hipkins has been a high-profile minister for the entirety of Labour’s government, taking on big portfolios including education, health, Covid-19 response and police.
He has largely avoided major scandal — with the notable exception of his handling of the Charlotte Bellis managed isolation and quarantine case — and his confidence in the face of media scrutiny has usually served him well.
Questioned yesterday afternoon, he would not rule out a tilt at the top job, but said it was too early for that kind of speculation.
"I’m not going to get into a discussion of that at the moment ... I think people need just a little bit of time," Mr Hipkins said.
"I think we will have a good, constructive discussion [and] we will select a new leader that the party will then unite behind."
Dr Woods is also seen as competent under pressure with a good grasp of detail in her hefty portfolios. These include energy and housing, which she took over after the KiwiBuild programme began falling short of its ambitious targets.
She said yesterday she would not add to speculation about likely leaders.
"Apart from the announcement that’s been made about Grant’s intentions, this is a discussion for Caucus and no-one will be ruling in or out."
Mr Wood has been something of a rising star in the party in his transport, immigration and workplace relations roles.
He would not speak about leadership prospects.
"There’ll be an internal caucus process and that’s all I’ll be saying today."
Ms Allan is a younger and more recent addition to Labour’s ministerial ranks, but has been spoken of alongside former whip Kieran McAnulty as a possible future leader.
However, Mr McAnulty ruled himself out of the running yesterday afternoon, saying there were no candidates yet but he "most certainly will not" be one of them.
Ms Allan would not make any kind of ruling over possible leaders.
"We believe it’s really important to stick to process here, so I think it’s really important that people don’t rule themselves in or out ... I’m not going to talk about the process. I’ll leave that for someone else," she said.
Of other Cabinet ministers, Minister of Foreign Affairs Nanaia Mahuta declined to say yesterday if she would put her name forward, saying it was a caucus process.
Broadcasting Minister and leader of the Maori caucus Willie Jackson said the Maori and Pacific caucuses would be meeting together, and there could be a “number” of MPs who put their names forward.
He “pretty much” ruled out putting himself forward.
