Starmer plans to use a speech on Monday to argue that he can change tack and revive his government's fortunes.
But his position is fragile as rivals weigh their options.Â
Labour MP Catherine West said she will try to trigger a leadership contest if she does not like what she hears in the speech.
Former deputy prime minister Angela Rayner, often seen as a potential challenger, said the party "needs to change" although she did not explicitly call for Starmer to go.
She said "the prime minister must now meet the moment and set out the change our country needs".
Starmer is trying to regain momentum after heavy losses for Labour in local elections across England and legislative votes in Scotland and Wales.
Last week's elections were widely seen as an unofficial referendum on Starmer, whose popularity has plummeted since he swept to power in a landslide less than two years ago.
His government has struggled to deliver promised economic growth, repair tattered public services and ease the cost of living, and been hamstrung by repeated missteps and policy U-turns on issues including welfare reform.
He has been further hurt by his disastrous decision to appoint Peter Mandelson, a scandal-tarnished friend of Jeffrey Epstein, as the United Kingdom's ambassador to the United States.
In last week's elections Labour lost votes to both the populist anti-immigration Reform UK and the "eco-populist" Green party.
It reflects the increasing fragmentation of UK politics, long dominated by Labour and the Conservatives.
Still, Starmer told The Observer newspaper on Sunday that he wants to remain in office for a decade.
He is pinning his hopes on Monday's speech and an ambitious set of legislative plans to be set out in a speech on Wednesday by King Charles III at the state opening of parliament.
None of the high-profile Labour politicians considered potential challengers to Starmer - including Rayner, Health Secretary Wes Streeting and Greater Manchester mayor Andy Burnham - has called for him to resign.
But a growing number of Labour MPs urged the prime minister to set a timetable for his departure.
UK politics allows parties to change leader mid-term without the need for a new election.
Josh Simons, a formerly loyal Labour MP, wrote in the Times of London that Starmer "has lost the country" and "should take control of the situation by overseeing an orderly transition to a new prime minister".
West, a former junior minister, is trying to hurry the process along, vowing to try to trigger a leadership contest unless Starmer delivers a barnstorming speech on Monday.
West acknowledged she does not have the support of 81 colleagues, needed to force a contest, and her move appeared to be an attempt to force more high-profile contenders to make a move.
"Working people sent us a message," West said.
"We have to listen to that, and we have to change and we have to do it quickly."
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