Pjanic played under Allegri between 2016 and 2019, during which time the Old Lady won six domestic trophies, and that success is believed to be a driving factor in the central midfielder wanting a return to Juve.
The 31-year-old has endured a frustrating season with Barcelona, making only 19 La Liga appearances, with just six of those as a starter for Ronald Koeman's side.
The Daily Mail reports that Pjanic considers Allegri to be like a father figure, and the Bosnia-Herzegovina international hopes to end his Barcelona career this summer with a return to Turin.
It is claimed that Allegri will want to rejuvenate the Juve squad this summer after the side lost their domestic title reign, but Pjanic's age at 31 may hinder his chances of a return to the club.
Pjanic made 178 appearances for Juventus during his first spell with the club, finding the back of the net 22 times and providing 41 assists.