Labour MPs will urge the Prime Minister to set out a timetable for his departure in the wake of the local elections.
Following what is expected to be bruising results, MPs are expected to write and call for beleaguered Keir Starmer to stand down by party conference in the autumn.
This would allow time for a contest to be held and a new leader to take over in an orderly manner.
Crucially, it would also buy time for rivals such as Andy Burnham to find a route back to Westminster and Angela Rayner to resolve her tax affairs.
It comes amid warnings that losses of more than 1,500 council seats on May 7 could trigger a Cabinet revolt.
Labour sources suggested that Mr Burnham's allies were behind the letter to oust the PM, and that an MP from the North-West could stand down to give him a seat.
Names of potential candidates include Marie Rimmer, Charlotte Nichols and Peter Dowd – though the former two said they had not been asked to make way for him.
The statement is said to be circulating among MPs and will be sent in the aftermath of torrid local election results.
Labour MPs will urge the Prime Minister to set out a timetable for his departure in the wake of the local elections Pictured: Sir Keir Starmer visited Kenston United Synagogue on Thursday
One source said: 'Most of the MPs on the Left of the party are turning against Angela or recognising her weaknesses are too great. So they're moving towards waiting for Andy.'
On Sunday allies of Mr Burnham said they were not aware of the move but said they had noticed that Labour MPs across the party were growing tired and wanted change.
MPs pointed out that should Sir Keir stand down ahead of party conference, it would allow time for Labour's ruling body to elect members who are close to Mr Burnham.
The current panel – which is packed with Sir Keir's allies – blocked the Manchester mayor from standing in the Gorton and Denton by-election.
However, some Labour MPs questioned why there was any talk of replacing the PM ahead of the local elections, saying any move should come afterwards to avoid handing political rivals a victory on May 7.
Another said the intent to remove the PM was more important than the process, saying: 'When the time comes and if the numbers are there, a process will be found.'
The threats to Sir Keir follows one of the worst weeks of his premiership during which his judgment in appointing Peter Mandelson was repeatedly questioned.
Sir Olly Robbins, the official he sacked over the vetting scandal, has claimed he was put under pressure to approve the appointment by No 10.
Labour sources suggested that Andy Burnham's allies were behind a letter to oust the PM, and that an MP from the North-West could stand down to give him a seat Pictured: Greater Manchester Mayor Mr Burnham arrived at Downing Street last month
This week, there will be further evidence from key figures involved in the appointment of Mandelson which could spell trouble for the Prime Minister.
He also has the shadow of a possible investigation by the Privileges Committee hanging over him. Several letters are believed to have been submitted to the Speaker who will decide whether MPs vote on the possibility tomorrow.
Sir Keir also faces a Cabinet revolt amid warnings that losses of more than 1,500 council seats on May 7 would provoke a challenge.
A loss on this scale would amount to more than half of the seats Labour is defending during the local elections, though some analysts suggest they could be greater than that.
'That would be the cut-off for a collective nervous breakdown among cabinet colleagues,' one Cabinet minister told the Sunday Times.
But a defiant Sir Keir vowed to fight the next election and insisted most of his MPs still backed him following one of his worst weeks in the job.
He said: 'What you never hear from are the all the people who are supportive, loyal and just want to get on with the job.'
However, he has faced opposition from several Cabinet ministers in the past week who have struggled to defend his actions over the Mandelson scandal.
These include Energy Secretary Ed Miliband, Foreign Secretary Yvette Cooper, Deputy PM David Lammy and Chief Secretary to the Prime Minister Darren Jones.
It came as polling found that 50 per cent of voters thought that Sir Keir should stand down, and 17 per cent favoured Mr Burnham as his successor in that scenario.
The majority, however, had no preference or didn't know, the Ipsos poll by Global showed.
A Labour source said on Sunday: 'Keir Starmer is one of only four Labour leaders ever to have won a general election. He has a clear five-year mandate from the British people to deliver change, and that is what he will do.'