Polling by YouGov suggested Andy Burnham would be Labour members’ preferred candidate for leader.
Sir Keir Starmer will assemble his new-look Cabinet for the first time on Tuesday after insisting again that he would not “walk away” from Downing Street.
Tuesday’s regular Cabinet meeting will be the first since Wes Streeting resigned as health secretary and called on the Prime Minister to quit last week.
It also follows Greater Manchester Mayor Andy Burnham’s announcement that he intends to fight the Makerfield by-election.
Mr Burnham has said a vote for him would be a vote to “change Labour”, while Sir Keir has insisted he will not set out a timetable for his departure even if the mayor returns to Parliament.
If Sir Keir does not resign, Mr Burnham would need to secure the backing of 81 Labour MPs to mount a leadership challenge, potentially setting up a contest with the Prime Minister for party members’ votes.
But on Monday evening, one of his supporters suggested the mayor would not seek an immediate leadership fight.
Leeds MP Alex Sobel told LBC it was not his “expectation” that Mr Burnham would immediately trigger a contest, saying he would “come back and potentially serve in the Government, try and help us turn this Government round”.
Meanwhile, polling by YouGov suggested Mr Burnham would be Labour members’ preferred candidate for leader, with 47% saying he would be their first choice for the job.
Some 31% ranked Sir Keir as their first choice, followed by former deputy prime minister Angela Rayner on 8%.
Just 4% ranked Mr Streeting as their first choice, with 57% saying he was wrong to resign as health secretary and 15% backing him in a head-to-head contest with the Prime Minister.
In a contest between Mr Burnham and Sir Keir, 59% said they would back the mayor while 37% would support the Prime Minister.
Amid continued leadership speculation, Mr Burnham is also likely to face further pressure to set out his stall for Labour members.
On Monday, the mayor said he wanted to see more power devolved to local areas, complaining of a “bloated national state” and a “malnourished local one”.
But he also appeared to row back on previous comments on the EU, saying he was “not proposing that the UK considers rejoining” the bloc and warning against rerunning the arguments of the Brexit years.
Mr Burnham has previously said he wanted to see the UK back in the EU within his lifetime.
