Andy Burnham has a credible plan to return to Westminster “within weeks”, his allies have said, with the Greater Manchester mayor expected to use a byelection fight to set out a new agenda for government.

Burnham, who was blocked by Labour’s ruling body from running in February’s Gorton and Denton byelection, has identified several seats where MPs are prepared to step aside for his leadership bid.

In a sign that his campaign is more progressed than previously thought, Burnham’s team is understood to have lined up an “impressive” candidate to replace him as Greater Manchester mayor.

Allies said he planned to outline a “radical rewiring” of the state in the coming weeks – including sweeping changes to the electoral system and a 10-year growth plan – after a potentially devastating set of elections on 7 May that could end Keir Starmer’s premiership.

After a fortnight that left Starmer fighting for his political future over the appointment of Peter Mandelson as US ambassador, the number of MPs backing Burnham is understood to have grown to far more than the 80 required to challenge the prime minister.

However, his supporters said they hoped to avoid a formal leadership challenge and to engineer a process where Starmer would set out a timetable to stand down soon after next week’s votes for the Scottish and Welsh parliaments and councils across England.

MPs have discussed the possibility of Burnham offering Starmer the chance to stay on as foreign secretary and continue work on the Iran war and Ukraine. Ed Miliband and Angela Rayner, another leadership rival, are expected to be offered top jobs in a Burnham government.

Burnham, who has made a series of policy speeches at ideologically aligned thinktanks in recent weeks, is said to be preparing an explicit programme for government that would be announced at a prospective parliamentary byelection campaign. Several possible seats have been identified in Greater Manchester and Merseyside.

Pledges backed by Burnham, a former health secretary, include introducing proportional representation across the UK, a 10-year plan for local services and an overhaul of inheritance tax to pay for the social care system.

It is understood that an “impressive” candidate, who is not a sitting MP, has been lined up to contest the mayoral election that would be triggered if he is allowed to stand for a parliamentary seat.

Bev Craig, the leader of Manchester city council, is seen as the frontrunner and has not ruled herself out of a contest. Craig declined to comment when contacted by the Guardian.

Backers of other potential leadership challengers – Wes Streeting and Rayner – are both said to have 80 MPs willing to back their candidates for an immediate challenge to Starmer.

But Burnham supporters said they hoped to convince the prime minister of the need for a more stable transition – rather than a bloody leadership contest – which would give the man dubbed “the king of the north” time to return.

“There are very strong possibilities of this happening within weeks but certainly months and over the summer,” one Burnham ally said.

Starmer is highly unlikely to play any part in facilitating Burnham’s return, and the two are not on good terms. Members of Labour’s national executive committee – which blocked Burnham from standing in Gorton and Denton – told the Guardian there was no route for the mayor through that committee.

Some allies of Burnham are believed to have been attempting to convince union general secretaries to change their views, with Unison the key target. Others, such as GMB, have ruled out supporting Burnham because of his closeness with Miliband, who is at odds with the union over oil and gas licences.

One Burnham ally said: “He’s just so obviously the person best placed to turn the Labour party around.

“He has popularity in terms of polling, in terms of personality and a plan – that will become more evident the other side of 7 May, when there will be more depth about how do you replicate and scale Manchesterism across the country. There will be a broader policy plan that will come out the other side [of the elections].”

But one MP backing Burnham said there was a lack of coordination between the disparate groups who might back him, including impatient modernisers and “red wall” or Blue Labour MPs.

“Andy’s big potential lies in the coalitions he can build,” another MP said. “His test will be whether he can build beyond a core of [the soft-left] Tribune group, who really don’t want Angela [Rayner].

“That kind of coalescing is what we need to show we’re different from the Tories, we don’t descend into months of bloodletting and chaos.”

One MP on the right of the party said: “I could get behind Andy but I need to know from him why someone like me should back him.”

Elections to the national executive committee, which some in Westminster have speculated could move things in Burnham’s favour, will not take effect until after Labour conference in the autumn and are unlikely to upset the balance significantly.

New polling suggests that just one in 10 voters believe Starmer should stay as prime minister after the May elections, if predictions are right that the party will lose more than 1,850 councillors – 75% of the seats it is defending.

The polling from Portland found Burnham was seen as the politician most likely to be “strong and decisive”, competent at getting things done and genuinely understanding the lives of ordinary people.

But the poll found people favoured Starmer as the politician with a more clear vision of the country and someone honest about trade-offs. The scores for each politician, however, were extremely low compared to the numbers polled who picked “none”.