Andy Burnham said he would seek to enter any potential Labour leadership contest should he win the Makerfield by-election on 18 June.
The Greater Manchester Mayor also told BBC Question Time that he would "take the fight to change politics and change this county as high as I could".
But Reform UK candidate Robert Kenyon said there is a need for a "big move away from career politicians" and warned against using Makerfield as a "stepping stone".
Prime Minister Sir Keir Starmer has defied calls from some Labour MPs to step down after poor election results last month. No formal challenge has been launched against him.
Sir Keir also said he would not "walk away" from Downing Street despite suffering ministerial resignations, most notably from former Health Secretary Wes Streeting.
The by-election was called after then-Labour MP Josh Simons stepped down, a move he said was to allow Burnham to run for Parliament and potentially mount a leadership bid.
Burnham and Kenyon, a plumber and local councillor, were among five of the 14 candidates in the by-election who appeared on Question Time in Makerfield.
Asked about how politicians could restore public trust in them, Burnham criticised "point-scoring before problem-solving" in Westminster before he was pressed about his own ambitions.
The Labour candidate said: "I can't do anything unless I'm lucky enough to get the support of people here.
"But if I get your support, I would seek to represent you at the highest possible level and give this constituency maximum power and influence.
"I think Wes Streeting seems to have launched a leadership contest, so if that is running I would seek to join it, but I'd have to persuade members of the Parliamentary Labour Party to do the same."
Any challenger wanting to trigger a leadership contest must be an MP and have the backing of 81 Labour MPs.
Streeting has confirmed he would enter any potential contest and has encouraged people to back Burnham in Makerfield as he wants a "proper contest with the best candidates on the field".
Kenyon criticised Burnham's record as mayor in Manchester saying while that city "thrives we're struggling to survive in Wigan", adding: "We've been left behind."
He said there was a need to "get normal people in politics, people who care about the place, people who live in the place and people who want to stay here and not use it as a stepping stone for other things".
Conservative candidate Michael Winstanley questioned why Burnham was seeking a return to Parliament after previously pledging to serve his current term as mayor, adding he was "disgusted" and "sad that we're having this by-election foisted upon us".
He added: "We will get rid of Keir Starmer whichever politician is elected in this by-election and if you vote for me, it'll speed it up straightaway."
Burnham replied by saying "you cannot control events in politics" as he highlighted Labour's losses to Reform in recent council elections in the area.
Green Party candidate Sarah Wakefield said there was a need for "serious conversations about who is contributing for a better future for our children, to solve the climate crisis, to get our high streets going, to make sure that we have the money back in towns and villages".
Liberal Democrat candidate Jake Austin said the by-election was an "election for a potential future prime minister via the backdoor and that's not the right way to be doing politics".
A full list of the parties who have put forward candidates for the Makerfield by-election can be found here.
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