A No 10 spokesperson said: “The prime minister will not walk away from the mandate he was given just two years ago to build a stronger, fairer Britain.”
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.
The vote in Makerfield is expected to be a race between Burnham and Kenyon, a plumber and local councillor.
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, saying while Manchester “thrives we’re struggling to survive in Wigan” before 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”.
