Sullivan has admitted that two or three players have indicated that they would like to leave Upton Park this summer and would be allowed to go if the club were offered an acceptable fee.
England striker Cole is one such player that could leave east London after a fee was agreed with Stoke over a potential move.
While West Ham wait for Cole's decision, Sullivan promises reinforcements will be brought in to replace outgoing stars.
"The ball is in Carlton's court," said Sullivan. "It is a question of whether he wants to go or whether he wants to stay.
"You're very much in the hands of players. Two or three players have indicated they would like to leave the club and if they get moves that are acceptable to the club then we cannot stand in their way.
"We will then bring in replacements, so a lot depends on who comes and who goes.
"I think people will expect to get them on the cheap but we will be realistic.
"We will not put valuations on players that are completely unrealistic.
"One or two players we will virtually give away who are on quite high wages and don't want to be at the club and don't think fit in with our formation.
"But other players, if we don't get a substantial fee, will be starting the next season with us."