Bill Shorten appears on track to take the federal Labor leadership at a caucus meeting in Canberra on Friday.
But members of Labor's Left are appealing to Anthony Albanese to contest the top job under new rules which give grassroots members a say.
The leadership battle looms as outgoing minister Brendan O'Connor said former prime minister Kevin Rudd should quit parliament to give the party some clear air in opposition.
"Kevin should seriously contemplate leaving the parliament to allow whoever's the leader to get on with us taking it up to the coalition," Mr O'Connor told Sky News on Tuesday.
"If you have a former prime minister sitting in your party room on the backbench, that spectre looms large."
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
Mr O'Connor, from the so-called Victorian "soft Left", said both Mr Shorten, from the Right, and Mr Albanese, from the Left, had the "virtues and qualities" for the top job.
It's understood the Right faction endorsed Mr Shorten on Monday, but it is not yet clear whether Mr Albanese will also put his name forward or seek to stay in his role as deputy.
A Labor Left source told AAP on Tuesday there was a "phenomenal groundswell of grassroots support" for Mr Albanese to put his hand up for the leadership.
"There is genuine enthusiasm amongst Labor's membership for him to take it on," the source said.
Mr Rudd will formally step aside at Friday's meeting after saying on election night on Saturday he would not recontest the leadership following a sweeping coalition victory.
Before the election, Mr Rudd convinced caucus to change the ALP rules so the parliamentary leader is chosen by a ballot weighted 50 per cent caucus members and 50 per cent grassroots members of the party.
If there are two nominations on Friday, the new ballot process will more than likely go ahead over several weeks.
But if there is only one nominee, he or she is expected to be endorsed on the day.
Under the rule change, the caucus will elect the executive but the new leader will determine what portfolios they hold.
On Monday night, former minister Craig Emerson also called on Mr Rudd to retire from parliament, saying the former prime minister would continue to "wilfully, recklessly, destabilise Labor leaders" as long as he remained in the party.
Rudd loyalist Kim Carr said Mr Rudd intended to stay for the full term, having been re-elected to the seat of Griffith on Saturday.
