The majority of Labour members say they do not believe Keir Starmer can turn around the party’s fortunes, while 45% say the prime minister should step down.

The mayor of Greater Manchester, Andy Burnham, was the first preference for 42% of members, who were asked to rank their preferred successor.

Several Labour MPs – especially those who are close to Burnham – have told the Guardian that they would like to see a timetable for Starmer to stand down in an orderly and dignified way, including allowing the mayor time to seek a parliamentary seat.

The poll was conducted just before Thursday’s elections, where Labour was fighting on all fronts, in local elections in England and parliamentary elections in Wales and Scotland. Nigel Farage’s Reform UK won hundreds of seats and control of more councils in England, Labour went backwards in Scotland as the SNP claimed a historic fifth victory and Plaid Cymru ended a century of Labour dominance in Wales, unseating the Labour first minister Eluned Morgan.

In London, the Greens took several councils across the capital, a sign of growing progressive disillusionment in Keir Starmer’s party.

Among members, the poll found Burnham has a net favourability of 72%, suggesting he would be likely to win any leadership contest that he was allowed to contest. Burnham cannot run while he is denied the chance to run for parliament, which has been blocked by Labour’s governing national executive committee (NEC).

The poll of more than 1,000 party members shows a sharp rise in the number of members dissatisfied with Starmer’s leadership – in October just 28% said the prime minister should resign if Labour suffered poor election results in May.

More than a third of members polled said they were considering resigning their membership, according to the polling by the thinktank Compass.

The thinktank is affiliated with Mainstream, a Labour caucus that is a prime backer of a potential Burnham leadership bid.

The polling found more than half of Labour members (51%) do not believe Starmer can turn around the polls – suggesting they do not have faith he could beat Reform at the next general election.

But Starmer is not the only target of members’ ire – 49% said the prime minister was jointly responsible for the party’s dire results, along with the rest of the national government. Of those polled, 36% had considered cancelling their membership.

Lena Swedlow, the deputy director of Compass, said: “Voters made their displeasure with Starmer’s leadership heard on Thursday and now Labour members have too. Nearly half want him to stand down and over half don’t think he’s the man for the job.

“In the face of an incredibly serious economic situation at home and abroad, it’s essential that the country and the party have a leader that they can trust. There is a clear desire for change in both the politics and personnel of this government. Making these changes in an orderly, measured, considered way will be crucial for the stability and prosperity of the country.”

Burnham’s backers in parliament hope that by heaping pressure on Starmer and senior party figures that the prime minister will set a timetable for departure and that Burnham would not be blocked from standing for parliament again.

The mayor attempted to stand at the Gorton and Denton byelection but was blocked by the NEC on the grounds that the party could not afford to lose a Greater Manchester mayoral byelection.

Members of the NEC have told the Guardian the position will not change unless Starmer’s mind is changed – which is highly unlikely.

At least 10 MPs have called for the prime minister to put in place a timetable for departure, including the former transport secretary Louise Haigh, the chair of the women and equalities select committee, Sarah Owen, and several MPs elected in 2024 including Jonathan Brash, Simon Opher and Connor Naismith.