Cole has been unable to make an impact at Liverpool this season following Rodgers' appointment after spending last term on loan at French side Lille.
The midfielder has yet to start a Premier League game under Rodgers and was not involved when Liverpool won 3-0 at QPR on Sunday.
Redknapp handed Cole his professional debut at West Ham United and remains a big fan of the 31-year-old as he looks to improve his QPR squad.
"I've got a lot of time for Joe, as a player and a person, he's a good lad to have around your football club, a good player," said Redknapp.
"He's interesting me. I'd like to bring him here if we could. We'll see what happens.
"The deal would have to be right for the club. We couldn't do a deal with Liverpool with what his contract is at Liverpool. We'd have to try to find a way of doing a deal with Liverpool, if they're willing to let him go.
"I'll have to speak with Brendan, I'll have a chat with him when the time's right and see what he has to say.
"I haven't spoken to Joe, I'm not allowed to speak to Joe. Are you trying to get me banned by the FA?
"I haven't spoken to Joe for two years. I nearly took him to Tottenham a couple of years ago but he went to Liverpool, that was last time I spoke to Joe. He's a great lad."
The former Tottenham boss is battling to keep QPR in the Premier League and has reiterated his view that signing players in the January transfer window is not easy.
He added: "January is so difficult. It's a tough time to find players. We're in a desperate situation, but we'll see what happens this week.
"I've got a list of names, obviously. But getting them isn't easy and it's all about getting value. Attitude and ability are important for us. We need players who will come in and make an impact."
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