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.
"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'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.
