The 23-year-old is currently way down the pecking order at White Hart Lane, with the likes of Scott Parker, Sandro and Moussa Dembele all available to manager Andre Villas-Boas in central midfield positions.
This has tempted ‘Arry, who coached the player during his time at Spurs, to prepare a January move as he looks to strengthen his squad.
Redknapp is keen to bolster his squad in a bid to keep the Super Hoops clear of relegation, and is thought to consider midfield additions to be key.
Livermore himself may be keen to make the switch in a bid to gain vital experience and force himself into AVB’s first-team plans.
Fulham are also thought to be interested, with Martin Jol looking to bring in fresh faces ahead of the second-half of the season.
This may push QPR into forcing through a deal, as they will not want to miss out on another target after watching Joe Cole move to West Ham.
Redknapp is also keen on veteran full-back Anthony Reveillere as he looks to address the club’s right-back issues.
The 33-year-old will be allowed to leave Lyon for the right price.