Gazzetta dello Sport report Sassuolo are among the sides interested in his services as he ponders where to stay at Torino or move on.
Injuries hampered Zaza last season as he made only 11 appearances in all competitions, failing to register a single goal.
He went to the Euros that summer but missed a penalty as Italy were eliminated by Germany in the quarter-finals.
He then moved to West Ham and Valencia on loan, eventually joining the latter before returning to Serie A with Torino in 2018.
Zaza, 31, previously played for Sassuolo from 2013 to 2015 before moving to Juve.