Former West Ham and Stoke City forward Arnautovic was linked with a move to United as part of the recruitment drive for Erik ten Hag's first season with the club.
The Red Devils reportedly considered a move for Arnautovic after Cristiano Ronaldo outlined his desire to leave Old Trafford, but the 33-year-old insists he is happy with his choice to remain in Italy.
"Manchester United tried to sign me several times and it was a difficult decision to stay in Italy, because my family wanted to return to England," Arnautovic said.
"For me, it's clear that the Red Devils remain at the top of the universe and Bologna, as a name, are not.
"But that's fine. Here everything is calm, you have time for yourself and no stress. This is only good for me."
Arnautovic has scored eight goals in 13 Serie A appearances this season, a tally only bettered by Napoli's Victor Osimhen (nine).