The Hammers will move from their current Upton Park home to the facility in Stratford for the start of the 2016-17 campaign.
The East Londoners last finished in the top six during the 1998-99 season, but Brady has insisted that European football is what the club will be aiming for at their new ground.
"Our vision has always been to be a top-six club regularly competing in European competition," BBC Sport quotes Brady as saying. "The stadium will give us one of the best supporter arenas in the world."
West Ham are eighth in the current Premier League standings.