The 29-year-old, whose contract with the Old Lady runs out at the end of the season, is set to exit Allianz Stadium in the summer, amid overtures for a deal extension.
Inter have emerged as frontrunners in the race for the Ghanaian's signature, with several reports indicating that a deal has already been agreed between the two parties.
However, according to Tuttosport, Milan have joined the race for the ex-Udinese star.
Having been reduced to a squad player over the years at Juventus, Asamoah has reportedly set his heart on a move away in his search of regular action.
Last summer, Asamoah came close to sealing a move to Galatasaray, but the move fell through as Juve failed to find a replacement.
The Old Lady are said to be reluctant to release the Ghanaian amid speculations that first-choice left-back Alex Sandro could leave at the end of the season.
Five Serie A titles, three Coppa Italia winner's medals and three Super Cup titles make up Asamoah’s trophy haul in Turin.