The Bologna striker Arnautovic had been heavily linked with a return to the Premier League, where he had performed admirably for Stoke and West Ham.
Manchester United decided to take a swing at luring him to Old Trafford with a bid of around £7million.
The offer was quickly rebuffed by the Italian club who would have been looking for a lot more to prise away their star man.
Bologna sporting director Marco Di Vaio claimed the forward was "priceless".
"We want to keep going with Arnautovic and keep him, he's at the centre of our project," Di Vaio told Italia 1.
"Bologna's wish now is to keep our current team and strengthen it in order to have an important season. Money is not the issue, Marko is priceless for us."
The backlash against the possible signing of the Austrian was loud and vociferous from Manchester United fans, former pros and media pundits.
BBC pundit Chris Sutton called the move a 'PR disaster' on 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," he said.
Now Manchester United have reportedly stepped away from the chase and are concentrating on other possible bin-fires to poke a stick at.
Manchester United have received a number of complaints from fans, and chief executive Richard Arnold has been emailed directly.
There have also been an allegation that Arnautovic used an anti-Albanian slur in an international match against North Macedonia, which the striker denied. He also denied allegations he had racially abused a opponent while playing for FC Twente in 2009.