Spurs failed to secure the England stopper's signature over the summer transfer window, with Trotters boss Owen Coyle revealing the only bids they received for the 25-year-old fell well short of their valuation.
But Redknapp has not given up his pursuit of Cahill, who is out of contract at the Reebok Stadium at the end of the season and could leave on a free transfer next summer.
He told the Sunday Mirror: "We were able to negotiate a deal with West Ham for (Scott) Parker, but we couldn't get a deal done for Cahill.
"That's not to say we won't try again in January. If the chairman can reach an agreement with Bolton I'd love to bring him here."
Another player to have been repeatedly linked with a move to White Hart Lane is former England captain David Beckham, who is out of contract with Major League Soccer side LA Galaxy in November.
The veteran midfielder recently revealed staying in California would be the better option for his family but he is yet to sign an extension in Los Angeles, with French giants Paris Saint German also thought to be chasing his signature.
Speculation suggests Beckham could return to the Premier League for a loan stint at Spurs during the MLS off-season, but Redknapp has reiterated there is no agreement in place.
He said: "I'm not aware of any agreement with David.
"I've always had the utmost admiration for him as a player and he's a fantastic role model.
"I wouldn't rule anything in or out. The biggest problem is accommodating him in the side because I have an abundance of riches in midfield."