Players Scott Parker, Carlton Cole and Mark Noble travelled to the Prime Minister's residence in Downing Street to present a joint bid with the London Borough of Newham to run the 516 million-pound stadium.
The East London Premier League club plans to reduce the capacity of the stadium, capable of hosting both football and athletics as well as concerts, from a capacity of 80,000 to 60,000 seats.
The Olympic Park Legacy Company (OPLC) had set an end of September deadline for bids from those wishing to lease the stadium.
"We believe this is the only bid that can deliver London's legacy commitment to the International Olympic Committee," said West Ham vice-chairman Karren Brady on the club's website.
"The last thing anyone wants is for the Olympic stadium to become a ghost of Olympics past. The only realistic solution is to make the stadium work for a Premier League football team and that should be West Ham United."
The club and Newham say their bid, which would retain most of the stadium structure, would also bring wider benefits for the local community, with a proposal to incorporate an Olympic visitor centre and football museum.
The club's current Upton Park ground would become part of a regeneration scheme to include new houses, shops and other facilities.
"Our plans will deliver upwards of two million visitors each year and provide a significant beacon in the East End that will fulfil the original London 2012 bid commitments," Newham mayor Robin Wales said.
