Tottenham attempted to sign Cahill, 25, on the final day of the summer transfer window but, although he is in the final year of his contract, Bolton Wanderers were unwilling to sell at the price offered.
Bolton boss Owen Coyle recently said he believed Arsenal would be among several clubs to reignite their interest in Cahill during the January window and Redknapp has now indicated that a fresh bid could be in the offing.
"We couldn't get a deal done for Cahill," he said in the Sunday Mirror. "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."
Meanwhile, Redknapp said he was unaware of any plans to bring in David Beckham, whose contract at Los Angeles Galaxy expires in November.
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."