The 33-year-old Austrian international was linked to a return to the English top flight by several reports in the Italian media on Saturday.
Arnautovic has scored 14 goals in 33 appearances for the Serie A side, prompting reported interest from both Juventus and Manchester United.
Arnautovic had been rumoured to be rejoining West Ham last season after leaving the club under a cloud following a big-money switch to China in 2019. West Ham manager David Moyes had got the best out of the Austrian in his time at the club, scoring 22 in 65 appearances.
La Gazetto dello Sport and Get Italian News reported: "Arnautovic is said to be quite fascinated by the idea of returning to the Premier League, where he spent a fair amount of time during stints at West Ham and Stoke City.
"As of now, Bologna's price tag is said to be high as they want to keep hold of the Austrian."
United manager Erik ten Hag will be actively looking to bolster his forward options after seeing his team stutter to an opening defeat to Brighton on Sunday.
Sky Italy reported on Sunday evening that the Red Devils have had a bid of 9million Euros for Arnautovic rejected.
But the Dutchman remained cagey after the 2-1 defeat to Graham Potter's Seagulls, saying: "I don't go into names because I think in this pre-season 250 players are involved or linked with Manchester United, so that's it.
"We have two strikers, maybe a third one, with Marcus Rashford as well," he added on Sky Sports.