The midfielder, who has scored once since his return from a serious knee injury in September, is understood to be chasing wages in excess of £100,000 a week.
But Wilkins said he had spoken to former West Ham player Cole numerous times and said he was "upbeat" about his future at the club he joined in 2003.
"He loves Chelsea," he added. "He wants to stay at Chelsea and I am sure that the club and his people will sort the situation out.
"We are desperate to keep Joe. I know the club are and we as a management group feel the same.
"He is a very valuable asset to us. It is a contractual situation that he and the club I am sure will sort out at a later stage."
Cole scored against Wolves in November
Wilkins dismissed suggestions that contract wrangles were affecting Cole's form, insisting that he was returning from a 10-month lay-off and adding that the player is still in contention for a call-up to England's World Cup squad.
"He was outstanding in the Carling Cup against Queens Park Rangers and then you find, when you have had a long lay off, that your form does drop.
"Joe has just had a dip in form. He has shown in certain games, when he has played, a lot more of that good form.
"He has got to get a little bit more consistency back and then there will not be a problem with him.
"I think he will give England something slightly different. He can play on both sides of the pitch and in different formations so Joe has a very good chance of being selected for England."
Chelsea face Stoke on Sunday in the quarter-finals of the FA Cup and Wilkins says the holders still see the competition as a major target.
"The FA Cup is massive," said Wilkins. "Thinking back to last May and the joy we experienced in the dressing room at Wembley. These guys play this game to win trophies and we want to win the trophy badly."
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