"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 the BBC.
"That was a very special player."
He did not name the player but British media reports have linked Real Madrid's Dutch striker Ruud van Nistelrooy to the east London club.
"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," added former Birmingham City owner Sullivan.
"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."
NO SENSE
Sullivan said on Tuesday, when he took control of the club with David Gold, that West Ham had debts of 110 million pounds and recognised it made no commercial sense to buy it.
"There's 50 million owed to banks, 40 million owed to other clubs," he said.
"There's not a penny to come in, they (the previous owners) have borrowed against the next two years of season-ticket money."
Sullivan has talked of signing two strikers and a defender by the end of the January transfer window to help West Ham escape relegation.
The club are 16th in the 20 team league, ahead of Wolverhampton Wanderers and Hull City on goal difference and just a point clear of second-bottom Bolton Wanderers.
"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," said Sullivan.
"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."
