Andy Burnham will push to become prime minister in time to address Labour’s autumn party conference in Liverpool, his supporters have said.

The Greater Manchester mayor cleared his first hurdle to becoming the candidate in the Makerfield byelection on Friday as Labour’s ruling body gave him permission to stand for the seat.

If confirmed as Labour’s choice, he will need to win a difficult byelection in the Greater Manchester seat in mid to late June, with support for Reform UK having surged in the area at the local elections.

With the byelection most likely to take place on 18 June, some of Burnham’s supporters believe he has a path to becoming prime minister before parliament breaks for the summer recess, but sources close the mayor’s campaign believe his preference is a longer timeline.

The prize for Burnham winning a leadership contest in the summer would be a victorious homecoming and what one supporter described as a “natural moment” for a transfer of power in the north-west city.

Another source close to the mayor said he did not want to get ahead of himself and was concentrating on the byelection fight ahead.

Burnham’s route back to parliament was opened up by the resignation of his ally Josh Simons on Thursday, after other contenders to replace Keir Starmer backed away from triggering a contest against the prime minister this week.

Lucy Powell, Labour’s deputy leader, who will be the political lead for the party in the byelection, told the Guardian she expected the byelection “to really galvanise people from right across the party and we’ll absolutely be throwing everything at it”.

She added: “Should he be our candidate, Andy’s story, Andy’s narrative, Andy’s connection to the place will absolutely be at the forefront of it all. He is very much what keeps a seat like that in play for us.”

She described it as a “straight fight between us and Reform and exactly the sort of seat – a community and a place that perhaps once had more pride in itself than it does today – that you know feels that mainstream politics has not been listening to it enough and I think that’s why someone like Andy is so popular in a place like Makerfield.”

Starmer may still put up a fight against being replaced in No 10, despite having lifted his previous objections to Burnham seeking to re-enter parliament after a catastrophic week in which more than 80 of his MPs called for him to quit.

Steve Reed, one of his closest allies in the cabinet, conceded on Friday that the prime minister was “unpopular” but insisted Labour should not repeat the same pattern of changing leaders that damaged the Conservatives.

Wes Streeting, who resigned as health secretary this week but stopped short of challenging Starmer, gave his backing to Burnham standing on Friday, saying: “We need our best players on the pitch. There is no doubt that Andy Burnham is one of them.”

It is still possible that he or other cabinet ministers could push for a full leadership contest rather than a Burnham coronation if he wins the byelection.

Burnham is expected to set out next week his vision of “Manchesterism” – a politics of making people feel rooted and invested in their local areas – and how it could apply to the whole country.

Analysis by Survation shared with the Guardian suggests Labour with Burnham as the candidate could narrowly beat Reform by 45% to 43% in Makerfield, compared with Labour with another candidate attracting 27% versus 53% for Reform. The results were extrapolated using modelling from official election results, census data and survey data from the recent Gorton and Denton byelection.

In something of a blow to Labour, the Green party on Friday pledged to stand in the seat, a move criticised by the party’s former leader Caroline Lucas.

The contest has the potential to be a symbolic battle for the future of the country, with Nigel Farage’s Reform saying it will be fighting for the seat as hard as possible. Reform’s candidate is expected to be selected by the party’s HQ and there is understood to be concern that the party will face competition for votes from a candidate put forward by Rupert Lowe’s rightwing party Restore Britain.

Neal Lawson, a close adviser to Burnham, said he believed Burnham had “unique” electoral appeal in that he “appeals to both Green voters and Reform voters, and no one else does it”.

He said: “He’s always been confident about any byelection in the north-west. I think he sees in the seat that he’s got an opportunity to take on Farage and take on the right – and win. This is a real change. It’s not cosmetic. This is deep political, democratic, economic, social change. I think he can win on that ticket and through his commitment to PR [proportional representation], to pluralism, to new politics, he can win over the Greens as well.”

Some MPs are extremely nervous about what could happen if Burnham loses the seat and expressed dismay that he had not engineered a potentially easier battle. One Labour MP described Makerfield as the “absolute ground zero of red wall Reform anger” and argued that if the local election results were replicated then Burnham as the candidate would need to take almost every vote off the Greens and Lib Dems, as well as winning additional votes off Reform.