Filippo Inzaghi wants to keep on-loan Chelsea midfielder Marco van Ginkel at AC Milan for another season.
Liverpool defender Kolo Toure is close to agreeing an extension to his contract which expires next month.
The 35-year-old was a free agent after leaving the Sardinian club, where he had been the first-choice keeper for three seasons, in June.
A statement published on Torino's official website welcomed Avramov to the Stadio Olimpico without disclosing the length of his contract.
The former Pescara and Fiorentina keeper follows Fabio Quagliarella to Torino. The 25-time Italy international striker joined Giampiero Ventura's side from Juventus for 3.5million euros on Friday.