Starmer's Labour Party recorded the worst losses of a governing party in municipal polls since 1995, prompting a growing number of his own MPs to call on him to quit.
To try to shore up his position in the party earlier on Saturday, he named two influential Labour grandees as advisers, former prime minister Gordon Brown and former deputy Labour leader Harriet Harman.
But just hours later, Labour MP Catherine West, a former minister, told BBC Radio that she wanted the cabinet to work out a plan to replace Starmer by Monday, or she would challenge him for the position herself.
"If ... there are no leadership hopefuls who come forward tomorrow, then Monday morning I will put my name forward to stand for the leader of the Labour Party," she said.
As the extent of the defeat emerged, more than 20 MPs publicly and privately called on Starmer to set out a timetable for his departure.
Asked whether he would stand down he told media that was not the right thing to do.
"I'm not going to walk away from this," he said earlier on Saturday.
Earlier on Saturday, West said on X that she was urging Starmer to quit: "I know I speak for more Labour people than just myself in wanting him to step aside as our Leader."
Another Labour MP, Clive Betts, told BBC Radio on Saturday that he wanted Starmer to step down "in the not too distant future".
A number of cabinet ministers said on Friday that they continued to support Starmer, who just under two years ago led Labour to a landslide general election victory, and an immediate challenge from the potential leadership rivals does not look straightforward.
Greater Manchester mayor Andy Burnham does not have the seat in parliament he needs to mount a challenge, and former deputy prime minister Angela Rayner has yet to fully resolve the tax issues that prompted her resignation from office last year.
Wes Streeting, currently health minister, is - like Starmer - tainted by the appointment of Peter Mandelson as the United Kingdom's ambassador to the United States.
Streeting was close to Mandelson, who was sacked over his ties to the late sex offender Jeffrey Epstein.
Any candidate wishing to make a challenge would need to secure the public support of 20 per cent of Labour members of parliament.
With Labour currently holding 403 seats, that equates to 81 backers.
West said she had 10 names behind her so far but her preferred option was that another candidate put themselves forward.
"I think there are several people who would like to do it who have been planning for months," she said.
Aiming to reset his leadership and win back party support, Starmer's office announced Brown, 75, and Harman, also 75, would join his team.
"They're vital to how we strengthen our country and take it forward and provide the opportunities that give people that hope for a better future," he said when asked whether figures from the past could help with his plan for the future to improve people's lives.
Brown will seek to drive new defence and security investment and hone relations with the European Union, to try to boost economic performance, while Harman will focus on tackling misogyny and violence against women and girls, creating economic opportunities.
