New United boss Erik ten Hag was considering a move for the 33-year-old Austria international, whom he worked with at Dutch side FC Twente.
But is understood United pulled out because of a combination of Bologna’s growing transfer fee demands and the negative reaction.
United reportedly received complaints from fans opposed to the move.
Ten Hag dismissed talk of potentially signing the former West Ham United and Stoke City forward after the 2-1 defeat by Brighton in United’s Premier League opener on Saturday.
Midfielder Christian Eriksen started the game at Old Trafford in a false nine role, while Marcus Rashford and Jadon Sancho operated in wider positions.
Striker Anthony Martial was ruled out by a hamstring injury and Cristiano Ronaldo was not deemed fit enough to start after playing for only 45 minutes during pre-season.
Arnautovic scored 43 goals in 184 games in the Premier League between 2013 and 2019.
BBC pundit Chris Sutton said news of United’s interest was “a PR disaster”.
“That’s madness,” he said on BBC Radio 5 Live’s Monday Night Club.
“I thought it was a spoof. It’s ridiculous. It’s a real wrong move in terms of the eyes of the supporters.”