The post is due to be created in May 2017 and is at present held on an interim basis by former police and crime commissioner and MP Tony Lloyd, who is also expected to try to become Labour’s candidate.
Ivan Lewis, Labour MP for Bury South, is another potential contender.
The elected mayor will oversee delivery of the devolution deal negotiated with the government across Greater Manchester’s 10 boroughs, but for some issues will need the agreement of borough leaders to implement policies.
Burnham said the post was "a cabinet-level job, which needs cabinet-level experience".
He served as health secretary in the last year of the Labour government, and then held the shadow role during most of the coalition period before being appointed shadow home secretary last year after his unsuccessful bid to become Labour leader.
Burnham has been MP for the Greater Manchester seat of Leigh since 2001.
One potential embarrassment may be that just before last year’s general election Burnham indicated he might halt government plans to devolve £6bn of NHS spending to Greater Manchester saying he had “real misgivings” about the scheme. The devolution was later agreed.
Speaking in a pre-election debate run by six health organisations, Burnham said: “I support the principle of devolution…but what is happening in Greater Manchester seems to have been drawn up in haste on the back of an envelope.”
Other powers to be devolved to Greater Manchester include local transport, a £300m housing fund, parts of skills training and employment support and control of investment through an ‘earn back’ funding arrangement if the region reaches economic growth targets.
