Greater Manchester Mayor Andy Burnham, former deputy prime minister Angela Rayner and Health Secretary Wes Streeting are thought to be the main potential challengers to Sir Keir.
Andy Burnham
Although he has made little secret of his leadership ambitions, Burnham cannot enter a contest without first becoming an MP.
He was blocked from doing so earlier this year by the NEC.
Refusing to allow Burnham to stand in the Gorton and Denton by-election, it said the cost to the party of a mayoral election to replace him would be “disproportionate”.
The mayor’s supporters are hoping a leadership contest can be delayed until he is able to return to Parliament.
Angela Rayner
Rayner is favoured by some on the left of the party.
In her first public comments since the elections, Rayner called for Labour to offer regional mayors more economic powers and raise the minimum wage. “We are in danger of becoming a party of the well-off, not working people,” she said.
Rayner resigned as Sir Keir’s deputy prime minister last September after it emerged she had failed to pay the correct amount of stamp duty on a £800,000 flat in Hove.
It has been widely reported that she is still waiting for the result of an HMRC investigation into the purchase, which could complicate any immediate leadership campaign.
Wes Streeting
The health secretary backed the prime minister following its election losses, but did not explicitly rule out standing in any contest.
Some who would like to see him replace Sir Keir think this might be his very best chance, before Andy Burnham can get back to Westminster.
Streeting has been linked to Peter Mandelson, whose appointment as British ambassador to the US – despite his links to convicted sex offender Jeffrey Epstein – led to questions being raised about Sir Keir’s judgement.