Andy Burnham, who is standing as Labour's candidate in the Greater Manchester constituency with the hope of returning to parliament to replace Sir Keir Starmer as prime minister, urged people to "vote to change politics".
Sir Keir on Wednesday said he was willing to offer the Mayor of Greater Manchester a "big" job in his government, should his rival win when Thursday's votes are counted.
But allies of the mayor said he was not interested, insisting "the benefit Andy has is the wind of change for not having been associated with the government's failings".
The prime minister has repeatedly insisted he has no intention of standing aside 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 past two days and total pledges running into six figures, sources said.
But Mr Burnham and his allies are reported to be hoping for a "coronation" in which Sir Keir stands aside, rather than a contest, which they believe would be damaging to Labour's brand.
Meanwhile, the Guardian reported several cabinet ministers have had to be talked out of resigning early by the mayor's camp to prevent the government descending into chaos ahead of the by-election.
Burnham supporters within the government are said to have intended to quit early to pressure Sir Keir into making way for his rival.
Fellow leadership hopeful and former health secretary Wes Streeting is reported to have met Mr Burnham while campaigning in Makerfield on Monday, though a source told the paper "no deals were done" as a result of the meeting.
Sir Keir has said he plans to call Mr Burnham after the weekend, but he may face swift pressure to accept a leadership contest or stand down, as Mr Streeting has indicated he is prepared to trigger a race as early as next week.
In a video posted to X just after polls opened, Mr Burnham said: "This is a change by-election.
"We left Westminster 10 years ago to build something different, something better.
"We worked on a place-first basis, rather than party first; problem-solving rather than point scoring.
"We've taken that fight for people here, and I'll carry that fight forward if I am lucky enough to be elected as the MP for Makerfield. I'll take that fight as high as I can possibly take it."
Mr Burnham's nearest competitor in Makerfield is Nigel Farage's Reform UK, which polls have suggested is losing voters to Restore Britain, a party to its right with a more hard-line approach to migration and other issues.
Mr Burnham on Wednesday urged voters not to back Reform's candidate Robert Kenyon, as he claimed the party's victory could lead to a "darker and more divided politics".
"There is a path that carries on and makes Britain look more and more like the United States of America, and we do not, in my view, want to end up there, where people can't talk to each other in the street. That's not the Britain I know, and the Britain that I love, and that's not the path that we should be taking," Mr Burnham said.
Makerfield is one of three parliamentary by-elections taking place on Thursday.
The other two are in the Scottish seats of Aberdeen South, and Arbroath and Broughty Ferry, where SNP MPs Stephen Gethins and Stephen Flynn have stood down after winning elections to the Scottish Parliament in May.
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