Opinion among Labour MPs is divided over whether Burnham should come back to the Commons.
In a social media post appearing to back Burnham, Jo White, MP and leader of the Red Wall group said: “A London stitch up will be a disaster for Labour.”
However, Graham Stringer, the MP for Blackley and Middleton South in Greater Manchester, said Burnham was “entitled to be ambitious but this isn’t the right opportunity”.
“I think it’s very high-risk. By-elections are a nightmare for any candidates standing on behalf of a sitting government. I don’t think Andy should take that risk.”
Labour won the Gorton and Denton seat at the 2024 election with a 13,000 majority. Reform UK came second with 5,000 seats narrowly beating the Greens candidate into third place.
The area has traditionally been considered a safe seat, but Labour’s drop in popularity since the general election means the party are not guaranteed a win.
As the prime minister’s popularity has declined, speculation has grown that he could face a leadership challenge after the local elections in May, at which Labour is expected to perform poorly.
Alongside Burnham, Rayner and Health Secretary Wes Streeting have also been touted as possible challengers.
Speaking to the BBC’s Political Thinking, Deputy Prime Minister David Lammy said: “I don’t know if Andy really wants to get back into Westminster politics.
“He may do, and there’s much merit to him.
“But I would caution those often around that start to destabilise the ship and say that we can have a new leader.
“No, we can’t. Because immediately you’d be forced to have an election.”