Wes Streeting has confirmed he would enter any potential Labour leadership contest, days after resigning as health secretary and telling the prime minister he had "lost confidence" in him.
Streeting said on Saturday: "We need a proper contest with the best candidates on the field, and I will be standing."
Meanwhile, Greater Manchester Mayor Andy Burnham told the BBC he was seeking to stand in the Makerfield by-election to "save" the Labour Party.
Burnham - who is widely expected to try to replace Sir Keir Starmer as leader should he be selected as the party's candidate in the by-election and win - said the vote must be a moment to "reclaim the Labour party, to save it from where it's been".
The prime minister is resisting calls to stand down and set a timetable for his departure following Labour's election losses in early May - and is expected to fight any challenge from likely contenders, including Burnham and Streeting.
Streeting resigned on Thursday but stopped short of formally launching a challenge to Sir Keir's leadership.
Asked on Saturday whether he had the backing of the 81 Labour MPs needed to trigger such a contest, Streeting said: "I do have support in the parliamentary party, but this week I also had a choice."
Speaking to reporters at a conference hosted by Labour-aligned political organisation Progress, he went on to say the party could have "rushed" into a contest.
But doing so without giving Burnham the chance to stand would mean a new leader would lack "legitimacy", he said - which would end up "extending the instability and uncertainty" in the party.
He said he would not speculate on Sir Keir's future, and that despite any "disagreements" between them the prime minister had "many remarkable qualities" and was "someone of enormous decency".
Asked what he would say to voters thinking about backing Burnham, Streeting said: "Vote for him, in Makerfield especially."
Burnham was cleared on Friday to stand in the by-election by Labour's ruling National Executive Committee, after the constituency's current MP said he would vacate it to make way for the Manchester mayor.
This would pave the way for him to return to Westminster as an MP - which he must do to join a leadership contest under Labour Party rules.
He told the BBC on Saturday that Labour "needs to be better".
"We've got to see this as a moment to reclaim the Labour party, to save it from where it's been - we can't just carry on as we are".
He said that he wanted Labour to "be part of working class people".
"I think Britain has been on the wrong path for 40 years, it started de-industrialisation, de-regulation of the buses, privatisation of life's essentials."
Asked whether the election was a vanity project and voters would be frustrated by another election, he said: "I think this is a very necessary election, it's about fixing politics because it's not been working for people."
Burnham said he would "put everything into it", adding: "We're going to change the conversation in this campaign. We're going to get Labour closer to these communities again."
The BBC understands the by-election is likely to take place on 18 June.
Sir Keir has faced mounting pressure from MPs to resign following elections on 7 May which saw Labour lose almost 1,500 councillors in England and suffer heavy losses in Wales and Scotland's national elections.
The BBC is aware of nearly 90 Labour MPs who have since called on Sir Keir to leave his post, or set out a timetable for his resignation.
However, more than 150 MPs have indicated support for the prime minister, or said it was not the right time for a leadership contest.
Sir Keir has told his cabinet that he will "get on with governing" and warned that a leadership contest could result in "chaos".
