Prime Minister Keir Starmer vowed to stay on as Britain's leader Monday, as another top aide quit and he prepared to face lawmakers furious that his government has become embroiled in the Jeffrey Epstein scandal.

Starmer is scrambling to shore up his premiership amid calls for his resignation over the appointment of Peter Mandelson as US ambassador, despite knowing he had maintained links to Epstein after the sex offender was convicted in 2008.

In an address to Downing Street staff, the embattled premier vowed to "go forward... with confidence as we continue changing the country", according to a government official speaking on condition of anonymity.

"The prime minister is concentrating on the job in hand," Starmer's official spokesman told reporters, insisting that the Labour leader was feeling "upbeat", despite increasing rumblings from members of parliament that his days are numbered.

In a fresh setback, Starmer's communications chief Tim Allan on Monday quit just months into the role, the day after his chief of staff, Morgan McSweeney, also resigned for advising Starmer to make the contentious Mandelson appointment.

The fallout from the appointment of Mandelson, sparked by emails showing that he remained friends with Epstein long after the latter's conviction in 2008, has grown into the most serious crisis of Starmer's turbulent time in power.

Several backbench Labour MPs, mostly from the left of the party who have never warmed to Starmer's centrist tilt, have suggested that the prime minister should follow McSweeney out the exit door.

UK newspapers have quoted senior ministers saying they think he will step down soon on condition of anonymity.

But a number of leading figures have defended him, as no clear successor has emerged while the party faces key local elections in May.

Cabinet minister Pat McFadden said he believed Starmer would still be leader this time next year, saying he had a five-year mandate. (AFP)