The Championship side had agreed to buy the venue, only to see the deal collapse over a legal dispute.
The Olympic Park Legacy Committee set a Friday deadline for new bids, and West Ham vice-chairman confirmed their intention to complete a simplified rental agreement.
Under the West Ham proposal, the stadium would keep its running track and be used as a multi-sport venue.
Brady said: "Having spent the last two years dedicated to this project I am privileged to once more be submitting West Ham United's bid to become the anchor concessionaire of the Olympic Stadium post the 2012 Games.
"From the outset it has been my firm, unwavering belief that the Stadium can truly become a multi-use destination of which east London and the nation as a whole can be proud. I have never lost sight of our vision to play our part, along with the Stadium's major stakeholders, in ensuring it grows into a global asset, the 'jewel in the crown' of the Park that will be watched by the world.
"Our vision for the stadium has always been about standing up for the promises made for London back in Singapore in 2005 and what they meant for our future generations. We are honoured to have once more received the support and backing of our friends at UK Athletics.
Joint-Chairmen David Sullivan and David Gold issued a statement saying: "We remain committed to this magnificent stadium and provided we are able to offer a world-class stadium for football for our prolific, loyal and passionate supporters we believe there is nobody better placed to help deliver the legacy for the East End community than us.
"This area - the one from which we both originate - deserves a true and lasting legacy to follow the 2012 Games. It will be a tremendous event that the whole world will be watching and will therefore present a real opportunity to showcase the UK at its best. However, the showcase must not stop there.
"We know that using the stadium as our home will bring with it huge responsibility, but we are fully committed to making it our home for at least the next 99 years. We will take on with pride the mantle of what that means in terms of giving something back to the community."