BILD says United have a long standing interest in signing Calhanoglu.
It is claimed that contact has already been made with the 26-year-old's representatives.
And United are said to be serious in negotiations regarding a potential free transfer next summer.
Calhanoglu is not expected to sign an extension at Milan, who will want to try and sell him in the January transfer window with his contract running down.
The Turkey international has been a regular in the league this season with Milan fighting at the top of Serie A after five games.
He predominantly plays through the middle but he has been used as a winger throughout his career.