Mr Burnham has ruled out calling a snap election if he enters Downing Street.
Andy Burnham has said he will challenge Sir Keir Starmer for the Labour leadership if he wins the Makerfield by-election.
The Greater Manchester Mayor has previously promised a vote for him in Makerfield would be a vote to “change Labour” but has not explicitly said he would launch a bid until now.
He is widely viewed as Sir Keir’s main rival for the top job if he wins.
Appearing on the BBC’s Question Time by-election special, he said he had asked his team to help “develop a policy” but will need the backing of the Parliamentary Labour Party.
“I’m not somebody who gets ahead of myself. I can’t do anything unless I’m lucky enough to get the support of people here,” he said.
“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.
“So that’s the only question… I’ve said to my team, let’s have a proper look at this and let’s develop a policy.”
Mr Burnham has ruled out calling a snap election if he enters Downing Street, a spokesperson said previously in response to speculation the former cabinet minister was considering such a move to give him a stronger mandate.
He is one of 14 candidates standing in the election, which was triggered when Labour’s Josh Simons stood down.
The vote is expected to be a race between Mr Burnham and Reform UK’s candidate Robert Kenyon, a plumber and local councillor.
Mr Kenyon has faced criticism over past social media comments unearthed by several newspapers, including his stance towards Brexit and abortion.
