The Premier League club has joined forces with the London Borough of Newham for a bid to take over the 537 million pounds stadium once the Games is over, outlining plans for an Olympic visitor centre and football museum.
"It's about realising the full potential of the Olympic Park," Brady, who recently joined the club along with new joint owners David Gold and David Sullivan, told the club's website.
"As well-established local organisations, Newham Council and West Ham United are best placed to make it happen.
"If achievable it is the ideal answer for those who rightly demand a sustainable legacy from the 2012 Games and not a white elephant."
Previously it seemed that Games organisers' commitment to an athletics track legacy at the stadium would put off prospective Premier League tenants but West Ham appear keen to move away from their nearby Upton Park stadium for a bigger venue.
"We acknowledge the need for the stadium to host world-class athletics and so it should," Brady said. "But it can accommodate football too - and a whole lot more. There has to be a way of achieving that."
The Olympic Park Legacy Company (OPLC), a public sector company set up to manage planning and development of the Olympic Park post Games, has invited organisations interested in operating in, or locating to the stadium after 2012, to come forward with expressions of interest.
Originally the 80,000-seater stadium was to be scaled back to 25,000 seats after the Games but the OPLC said in November that it had not ruled anything in or out while it has also been provisionally included in England's 2018 World Cup bid.
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