The site will remain in public ownership after negotiations between the club and the Olympic Park Legacy Company came to an end, allowing others to apply to become tenants after the 2012 Games.
"We would welcome a move by OPLC and government to end that uncertainty and allow a football and athletics stadium to be in place by 2014 under a new process. If the speculation is true, West Ham will look to become a tenant of the stadium while Newham will aim to help deliver the legacy," said West Ham vice-chairman Karren Brady and Kim Bromley-Derry, chief executive of London Borough of Newham in a joint statement.
"Our bid is the only one that will secure the sporting and community legacy promise of the Olympic Stadium - an amazing year-round home for football, athletics and community events of which the nation could be proud.
"The true legacy of London 2012 will be the creation of jobs and a generation of young people inspired by sport based around a community home for all by 2014. We remain committed to help deliver that legacy promise to the people of London and the nation."
OPLC's talks to sell to West Ham ended amid a legal dispute from Tottenham Hotspur and Leyton Orient over the decision of the Hammers' local authority to provide a £40m loan to fund the move from Upton Park.