United were also made aware of increasing fan backlash against the move, stemming from accusations Arnautovic used racist language during Austria's game against North Macedonia at the 2020 European Championship.
Arnautovic, who is of Serbian descent, later apologised and insisted he is not racist, although he was banned by UEFA for one game for his on-field outburst.
Sources added that United initially targeted Arnautovic because of a belief the 33-year-old could be signed for a relatively low fee, but Bologna's valuation of the forward has increased since initial conversations took place.
Manager Erik ten Hag is keen to add a physical back-up striker to his squad and worked with Arnautovic during his time managing FC Twente in the Netherlands.
Martial is injured and Ronaldo is not yet match-fit after missing most of preseason because of a family issue, forcing Ten Hag to start with Christian Eriksen as a False No.9 for the opening-day defeat to Brighton.
Sources have told ESPN United have been offered the chance to sign former Chelsea striker Alvaro Morata.