Senior cabinet ministers are urging the party to hand Andy Burnham an uncontested path to the Labour leadership should he triumph over Reform UK in the forthcoming Makerfield by-election.
Labour Party members have cautioned that a leadership bid from Wes Streeting would plunge the party into prolonged internal conflict.
"The alternative is months of factional warfare which will create a schism in the parliamentary party that cannot be repaired," one cabinet minister warned.
Senior figures are pressing both potential rivals to negotiate an arrangement that would spare the party a damaging contest.
One minister insisted Mr Streeting stood no realistic prospect of defeating Mr Burnham following a Makerfield victory: "If Andy wins Makerfield, he will be carried aloft into the Westminster tea rooms on the shoulders of Labour MPs."
The Prime Minister is facing mounting pressure from cabinet allies to step aside either before or immediately after the by-election result is announced.
Sir Keir Starmer was reportedly weighing his options over the weekend, having privately conceded he would be unable to fend off a challenge from the Greater Manchester mayor should Mr Burnham secure victory.
Labour's national executive committee gave the green light to Mr Burnham's candidacy on Friday, marking a significant shift from its earlier decision to block him from contesting the Gorton & Denton by-election.
Senior Labour members have said that Andy Burnham should not be opposed if he is elected in Makerfield
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A committee source explained the change: "It's a different circumstance and the NEC works on blocs. They have shifted and it's as simple as that."
Labour held Makerfield in 2024 with a 5,400-vote majority.
Mr Streeting publicly endorsed Mr Burnham's selection on Friday, declaring he represented Labour's strongest chance of holding the seat: "The Makerfield by-election will be tough. Votes will need to be earned. We need our best players on the pitch. There is no doubt that Andy Burnham is one of them."
Even allies of the Health Secretary have acknowledged the political reality. One Mr Streeting supporter conceded: "I disagree with pretty much everything Andy now apparently claims to stand for. But if he wins Makerfield, I'm not going to pretend there should be a contest. He will walk into No 10."
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Wes Streeting would lose in a head-to-head with Andy Burnham, Labour MPs have claimed
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The campaign is expected to last four weeks, with June 18 the earliest possible polling day.
Reform UK dominated last week's local elections in the constituency, capturing more than half the vote against Labour's 23 per cent.
Financial markets responded sharply to the prospect of a Mr Burnham premiership on Friday, with government borrowing costs climbing to levels not seen since 1998.
The Ftse 100 shed nearly two per cent, with banking and utility stocks bearing the brunt of the sell-off.
Water company shares tumbled by approximately eight per cent, while National Grid and SSE experienced comparable declines amid concerns over Mr Burnham's previous support for renationalisation.
NatWest, Barclays and HSBC all dropped around three per cent on fears of increased banking levies.
Mark Dowding, chief investment officer at RBC BlueBay Asset Management, said: "It is questionable whether investors would be prepared to give a candidate such as Burnham the benefit of the doubt, happy to sell first and ask questions later."
Simon French, chief economist at Panmure Liberum, attributed the turbulence to uncertainty over who might replace Rachel Reeves as Chancellor.
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