Mr Burnham told the Guardian the role was "a cabinet-level job, which needs cabinet-level experience".
The Leigh MP said Westminster had "become a bit of an irrelevance for some people" adding that "we really need to change the way politics works".
The first elections for the role, currently filled by interim mayor Tony Lloyd, will be held in 2017.
Mr Lloyd and another former Labour minister, Ivan Lewis, are also seeking Labour's nomination.
Mr Burnham, the former health secretary, had been expected to announce his candidacy on Thursday but his intentions became clear when one of his Twitter profiles was rebranded as "Andy4Manchester".
As a national political figure over many years, Andy Burnham will be seen as the first genuine big hitter to throw his name in the hat to become Greater Manchester mayor.
The Leigh MP's move will be read by many as him giving up on a Labour return to government in 2020, but it's also a sign of the shift in power from Westminster to the North West.
Mr Burnham wasn't part of any negotiation for devolution to Greater Manchester and he was completely taken by surprise by the deal to merge health and social care, claiming the area was taking responsibility for an "NHS funding crisis" rather than anything else.
But time passes quickly in politics and Mr Burnham now sees opportunity where he once predicted disaster.
He'll be up against two well-established candidates for Labour who have spent months visiting local party groups to garner support.
He may find that being a big player in the national game doesn't particularly endear you to those who've stayed home to hold the fort but he's a formidable addition to the panel Labour members will eventually choose from.
The race to be the mayor of Greater Manchester has just got a bit more interesting.
Powerful regional mayors are a key part of the government's drive to devolve more responsibility to local authorities for Whitehall.
About 2.7 million people live in Greater Manchester, which has been at the forefront of the devolution project under the "Northern Powerhouse" banner.
Among the new powers transferred to the new mayor - who will also hold the office of police and crime commissioner - are responsibility for a Â£300m housing investment fund and control of transport budgets.
The leaders of the 10 councils that make up the Greater Manchester Combined Authority will form part of the new mayor's cabinet.
Another shadow cabinet minister Luciana Berger, recently said she was considering running for the new "metro mayor" role in the Liverpool area.
Speaking to The Guardian, Mr Burnham rejected the suggestion his decision showed he did not think Labour could win the 2020 general election, saying the party could "gain strength" from the powers being given to cities.
Shadow home secretary Andy Burnham will put himself forward to become the mayor of Greater Manchester.