The fullback is about to enter new contract talks with Saints amid interest from Leicester City.
However, TMW says Milan are also keen.
Incoming coach Ralf Rangnick has recommended a move for Bertrand during talks with Milan and former Arsenal chief exec Ivan Gazidis.
Indeed, it's claimed Rangnick had hoped to sign him when in charge of RB Leipzig.
Bertrand has a deal with Saints to 2021.