The Hoops have been linked with a number of targets this January, with names such as Robbie Keane, Tim Cahill and Wesley Sneijder being touted.
However, Livermore is the only player Redknapp today deemed a concrete “possibility”, suggesting he would like to sign the midfielder if Tottenham was willing to sell him.
"Jake's a good player. There's lots of time for him,” Redknapp told reporters. “He's a possibility. He's a player I like. If Tottenham are willing to do something he's someone who'd interest me.”
Redknapp also responded to questions about MLS duo Keane and Cahill, though the 65-year-old admitted he was not close to signing either player.
He continued: "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."
The QPR boss also praised West Ham midfielder Mohamed Diame, but dismissed the notion of the player leaving Upton Park.
He said: "He's a top player. You'd love to have him here, but he belongs to West Ham."
The final target Redknapp discussed was that of Dutch international Wesley Sneijder, although the Rs' manager suggested the chances of a move materializing were remote.
Redknapp added: "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."