According to the Tuttosport newspaper, the Italian giants will make an early move for the 25-year-old in order to stay ahead of the competition.
Gündoğan, who signed a contract extension with the Bundesliga club over the summer has been heavily linked with a move away from the club.
Gündoğan recently played down reports linking him with a move away from Borussia Dortmund but opted not to rule them out completely.
“About my future? I currently have no idea. This is very, very far away,” Gündoğan told Sport1.
Gündoğan has started the season well in the Bundesliga under Dortmund’s new manager Thomas Tuchel. Dortmund sit in second place in the Bundesliga with 29 points in 12 games.
The 25-year-old has provided five assists and scored two goals in 21 appearances for Dortmund this season.