Saville is highly rated at Chelsea, but he is well down the pecking order in Jose Mourinho's star-studded side.
Chelsea are thought to be open to offers for Saville and the news has alerted a number of Championship clubs to his availability.
Saville enjoyed a successful spell on loan at Brentford last season, making 40 appearances for the Bees as he played a key role in helping the club win promotion to the Championship.
Brentford are thought to be keen on bringing Saville back to Griffin Park, while former Bees boss Uwe Rosler has seen a couple of bids to bring the midfielder to Wigan rejected.
Wolves are also keen on Saville, but Middlesbrough are thought to be leading the chase for the all-action midfielder due to the good links they have with Chelsea, having taken several of their younger players on loan in the last year.
Saville is understood to have held talks with Middlesbrough earlier this week as he weighs up his options with Aitor Karanka hoping to win the race for his signature.