"We've offered £100,000-a-week to a player today and we are still not sure we are going to get him," Sullivan told Radio Five Live.
"That was a very special player.
"There's no transfer fee involved and he's got the choice of almost every club in Europe. He has played at the highest level.
"He's down to three clubs and West Ham are the only English club still in the hunt for him. Whether we get him is another thing."
Sullivan did not mention the player involved, however, it is believed to be former Manchester United striker Van Nistelrooy.
The 33 year-old has been linked with a move to Upton Park, with Tottenham and Stoke also interested in the Dutchman.
Sullivan admitted West Ham, who are 16th in the Premier League, are desperate for a player who can make an immediate impact.
He said: "We can carry one exceptional player, who would make a difference on that wage, but generally we have to bring the wages down and in the summer we would hope to sign younger players on a fraction of those wages.
"We have looked at a few players in the Championship but the reality is that it is no good if they make the grade in 12 months' time, we need them to make the grade now.
"We have a crisis and you have to have a different strategy to what our long-term strategy will be."
He added: "We are trying to sign players because unfortunately we have come in very, very late and we have got a very unbalanced squad.
"We are particularly short of strikers and they are the hardest and most difficult position to fill. That's the short-term objective.
"We have guaranteed that we will not sell a player in the transfer window and we are looking at targets. We are doing our best for West Ham but at the moment we are drawing blanks."