While Arsenal boss Arsene Wenger has admitted that he is a big admirer of the playmaker.
Cole, 28, could not agree terms on a new contract with the Blues and will be released - along with Germany captain Michael Ballack and Brazilian Juliano Belletti - when the pair's contracts expire at the end of June.
Arsenal, Tottenham and Manchester United have all been strongly linked with moves for the England international, who insisted on Wednesday that he would not be discussing his future until after the World Cup.
NEWS:Cole concentrating on England
But Redknapp, who handed Cole his first team debut at West Ham, revealed that the club had made initial enquiries about securing the player's signature.
Speaking on talkSPORT, Redknapp said that Spurs had been in touch with Cole's agent about a possible move to White Hart Lane.
“I don’t know. It’s not impossible. I think that Daniel [Levy], the chairman, has spoken to Joe’s agent and he said that Joe didn’t want to discuss any moves until after the World Cup. Then he’s going to look at his options."
Redknapp had previously claimed that Cole had already done a deal with another club, but the former Pompey boss now believes he's in with a chance to sign the England midfielder.
He added: “He hasn’t decided anything yet and he hasn’t done anything definite. So there’s a chance. I suppose it all depends on the financial side of it and everything else, but according to his agent he’s not made up his mind yet.”
GEAR:Get your Spurs shirt for new season
Arsenal continue to be linked with a move for Cole, with the Gunners expected to eventually lose their battle to keep captain Cesc Fabregas at the club amid sustained interest from Spanish champions Barcelona.
Emirates Stadium supremo Wenger is rumoured to be interested in Barca enforcer Yaya Toure and Bordeaux star Yoann Gourcuff as new recruits should Fabregas move on
And he admitted that he is a big fan of Cole, whose availability on a free transfer would undoubtedly appeal to Wenger, who is not known for splashing the cash.
"I like him as a player because when he played against us he always did a lot of damage against Arsenal," Wenger told Sky Sports News.
"He can create impact through his creative play. He can dribble, he can pass, he can score goals."
By Saad Noor
‘England have the players to win the World Cup – it’ll be tough for Thomas Tuchel to do a bad job, with the squad he has at his disposal’ Former Three Lions winger backs new boss after gentle qualifying draw
‘This is the hardest Manchester derby to call in years – Manchester United shouldn’t be petrified, it feels like Manchester City aren’t in control of any game at the moment’ Former Red Devils striker thinks a surprise result could be possible
