Cole worked under Redknapp at former team West Ham, and the new chief in west London has admitted that he still thinks the player is talented despite a disappointing time at Anfield.
"I like Joe, I had him as an 11-year-old," Redknapp told reporters.
"The talent is there and I think there's still a top player in there. I haven't made any enquiries yet - I haven't made any enquiries for anyone."
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But despite the speculation, the 65-year-old has reiterated that the side's current objectives are to turn performances around, as they look to get off the foot of the table.
He added: "I was interested in Remy but that was then. It's a different situation now. He would be expensive. It depends on where we are in January.
"I've never discussed players with anyone. Here we are, we need to win matches now. January is way off."