Cole, 29, left Chelsea after the expiry of his contract last summer and, despite Spurs making clear their interest, he opted to join Liverpool. However, he endured a disappointing season and, with Kenny Dalglish now having a surfeit of options in midfield, it appears he will be allowed to leave the club.
With Spurs currently short on midfielders, Redknapp has revealed he would be willing to take a chance on Cole this season.
"I like Joe," he said. "I signed him as an 11-year-old. He is a good player. I've got all the time in the world for him. We haven't made any enquiries about him but I wouldn't sit here and say I wouldn't have him at the club because if I did I'd be lying. I do like him."
Redknapp indicated that a move for Cole should not be taken as a sign that Luka Modric will be allowed to depart for Chelsea.
"We want him here," Redknapp said. "The chairman, I think, will sit down with him in the next few weeks and have a chat with him about his contract. We want to keep him - the chairman made that plain at the start and he's not gone away from that.
"Yesterday when I spoke to him he said that Luka definitely wasn't going and that was the end of the matter as far as he was concerned. I keep seeing that the chairman is trying to push the price up. He is not trying to push the price up. He is not for sale. He has made that clear. He doesn't want to sell him."
Spurs are also keen to tie up a move for Manchester City forward Emmanuel Adebayor but, despite reports on Wednesday that the deal was done, Redknapp said: "I'm not sure. The chairman's dealing with that so I wouldn't honestly know where it's at. Whether it's done, nearly done, not going to be done - I honestly don't know."