Energy Secretary Ed Miliband reportedly made the suggestion in a private meeting about two weeks ago as Labour awaited Thursday's local election results.
The move reflects fears heavy losses could trigger pressure on the prime minister to quit, with party tensions rising before the scale of any setback is clear.
Potential successors including Angela Rayner and Wes Streeting are said to be preparing leadership bids if the elections prove especially damaging for Labour.
With Labour facing a historic wipeout, can any successor truly reunite the party and win back disillusioned voters?
Is the surge of Reform UK and the Greens signaling the permanent end of Britain's two-party system?
As political turmoil spooks markets, is the UK economy vulnerable to another Liz Truss-style financial crisis?