The Hammers have been handed the keys to the London 2012 centrepiece after completing long and tortuous negotiations with the London Legacy Development Corporation. The agreement sees the Barclays Premier League club become anchor tenants under a 99-year lease, with the stadium to be transformed into a 54,000-seater venue in time for the 2016-17 season.
"Hand on heart, it is very difficult to compete with Chelsea and Man City when they have external benefactors," Sullivan said. "Man United are a very special club and it is very hard to match those few clubs and Arsenal have got the Emirates so it takes a bit of doing. But over time we hope to close that gap and our aim is finish third or fourth in the table."
He added: "We realise we have to buy better, spend more money, but it is a gradual build-up.
"We've always said if the king of Saudi Arabia wants to come along, we'll step aside. Or if another person wants to come and join us and three of us put the money in.
"We accept it will take time to build the team up but you need a bit of luck with signings.
"I am seeing somebody on Monday and if we sign the player he would be the most expensive player the club has ever signed. We lack goals but I accept it takes time.
"West Ham is a community club, you've got to remember that. It's not all about winning with West Ham. It is about giving back, letting people in at affordable prices and playing the right way."
Sullivan suggested the player he is meeting could cost up to £14million and predicted the club could spend as much as £30million to £40million on a single player in as little as three years.
"I hope we get one this year and over the years we get more," he said of marquee additions. "Once we've moved, there will be more money available."