Mr Burnham’s victory sets up a potential Labour leadership contest unless British prime minister Mr Starmer agrees to step down.

Mr Starmer has insisted he will fight any attempt to challenge him and has been building up a war chest for a leadership campaign.

But Mr Burnham’s supporters urged the Mr Starmer to set out an “orderly and managed transition” of power.

Former cabinet minister Louise Haigh said she hoped Mr Starmer will “do what’s best for both the country and the Labour Party”.

Mr Burnham said Labour had a “final chance to change” after his decisive win in the Makerfield by-election.

In his victory speech, he urged his party to act now, saying there would be no second chance.

He said: “Everyone knows that politics isn’t working.

“Everyone can feel that the country isn’t where it should be. Tonight could, just could, be the turning point.” 

In a direct message to Labour MPs he said: “I do say to my own party: this is a final chance to change.

“This is what people said directly to me on the hundreds of doorsteps that I stood on. We must hear it, we must act upon it and we must get it right. There will be no second chance.” 

Mr Burnham gave up the Greater Manchester mayoralty by becoming Makerfield MP, winning the seat that was vacated by Josh Simons in order to allow him the chance of returning to Westminster and seek to become prime minister.

In an attempt at addressing the assertion that he was only seeking to become Makerfield’s MP to further his own ambitions, he said: “It will never be a stepping stone to me, but instead will be my touchstone.

“A Makerfield test at the heart of British politics will make sure that the places Westminster has neglected will now get fairness.” 

Cabinet minister Lisa Nandy said she wanted Mr Burnham “back at the top table”.

The British culture secretary said: “What Andy’s shown here is that there is something that he brings, a willingness to go out and fight for the change that people need, to take on any system and any person who stands in the way and to be bold and to wear his heart on his sleeve, and people have responded.

“I think that with him back in the top team, at the top table, helping to drive that change, I think we’ll be in a really strong position.” Mr Burnham has indicated he will not accept a job in Sir Keir’s Government.

He is not expected to launch a leadership challenge immediately, instead hoping Mr Starmer will conclude that he has to give up the keys to No 10.

Mr Burnham’s supporters believe the scale of his victory will put increased pressure on Mr Starmer to stand down.

But Mr Starmer has repeatedly insisted he has no intention of walking away from No 10 and is understood to have amassed a war chest to fund his campaign to fight any leadership challenge, as first reported by The Times.

He has the backing of a group of private donors, with fundraising having ramped up in the last two days and total pledges running into six figures, sources said.

Mr Burnham has indicated he would join a leadership contest if one was triggered, something that would require 81 MPs to line up behind a candidate.

Former British health secretary Wes Streeting has suggested he would be willing to fire the starting gun for a contest if Mr Starmer does not stand down.