Redknapp was presented with a list of potential targets ranging from Mohamed Diame to Wesley Sneijder when he fielded questions from the press on his plans for January.
The Rs boss was happy to confirm loan moves for MLS duo Keane and Cahill are under consideration, but it was Tottenham midfielder Livermore's name which brought the most revealing response.
He said: "Jake's a good player. There's lots of time for him. He's a possibility. He's a player I like. If Tottenham are willing to do something he's someone who'd interest me.
"To bring someone in like Robbie or Tim Cahill - you're taking them for about six weeks then you lose them at a time when you've got 10, 12 games to try to stay up. It's difficult.
"They'd improve any team, those two - they're fantastic professionals and great players. But I'm not close to doing anything with them."
West Ham midfielder Diame has a £3.5m release clause in his contract, and Redknapp said: "He's a top player. You'd love to have him here, but he belongs to West Ham."
But the Rs boss admitted any prospect of landing Inter's Sneijder is fanciful, saying: "I can't see that one coming off. I was offered him by an agent, but I'll take it with a pinch of salt."