The Turkish newspaper Haberturk have stated that the Merseyside club have been monitoring Medjani in recent months and are preparing a €3 million bid for the 30-year-old.
Medjani was signed to Liverpool as a youth from AS Saint Etienne back in 2003. The versatile player left the club after several loan spells in France.
The report goes onto state that Medjani is keen on a move back to the Premier League.
The Algerian international joined Trabzonspor from Monaco on a free transfer at the start of the season.
Medjani has previously played for Monaco, Valenciennes, Olympiacos, Ajaccio, FC Lorient, Liverpool, Metz and Saint Etienne.
The experienced defender has one assist and eight goals in 41 appearances for Trabzonspor this season.
Trabzonspor are currently in sixth place in the league and could move up to forth if they defeat Sivasspor this weekend.