Sullivan stressed that the Hammers and manager Sam Allardyce are working hard to strengthen the current squad but explained their hands are tied by the new Premier League financial regulations.
"Sam and I are working tirelessly on identifying new targets to add to what has been a very good window for West Ham United thus far," said Sullivan on the club's website.
"However, the new rules state that if a team exceeds the new wage limit they could be hit with a points deduction. As we are now close to that limit, players can only be brought in if there is money left in the wage budget, meaning any new signings at West Ham are dependent on other players leaving the club.
"I have seen the suggestion that we should just give away or sell a few fringe players to free up the wages but I must stress this is easier said than done and despite our hard work over the summer, it just hasn't been possible yet.
"I'm hopeful that things will change over the last seven days of the window and I can assure fans that we will be working hard - right until the last minute if needs be - to make sure we give West Ham United the best chance possible of topping last season's tenth-placed finish."