Burnham is running for a vacant seat in parliament and, if successful, would then be able to launch a leadership challenge to Starmer, whose authority has been badly damaged after poor local election results and other missteps since taking power, Reuters reported.
His spokesperson said Burnham had no plans to call an immediate election if he did become prime minister. Britain is next due to hold a national parliamentary election in 2029.
On Saturday, the Sun newspaper had reported that Burnham might call an early election should he ultimately replace Starmer as leader.