According to John Richardson of the Sunday Mirror, the Red Devils were interested in the Austria international in the summer and will go back in for him again in mid-season.
"It’s understood the 29-year-old's advisors have been made aware of the United interest and would be open to a move to Old Trafford," continued Richardson. "[Jose] Mourinho sees Arnautovic as a catalyst for some of his younger players, who he has criticised for a lack of maturity."
Richardson added that Arnautovic would cost in the region of £50 million and Mourinho is reportedly confident the forward would be able to fire the team into the top four.
The West Ham man's future has been a hot topic recently, with the player's brother saying it is "possible" the striker will move on soon as he wants to "play for a top team."
The Austrian previously worked with Mourinho during a spell with Inter Milan, where Arnautovic spent a season on loan in 2009-2010. Since then he's represented Werder Bremen and Stoke City before playing arguably the best football of his career at the London Stadium.
Arnautovic's improvement in form has been in accordance with a change in position. Former Hammers boss David Moyes moved him to the point of the team's attack and the West Ham man has relished the responsibility in the main, giving many defenders a torrid time in the Premier League.
When the tweak in his role was made, Bleacher Report's Sam Tighe was impressed immediately:
Despite only being at the club for just over a season, Arnautovic has already established himself as one of West Ham's top goalscorers in recent years:
Arnautovic does have all the tools required to be a complete centre-forward. Physically there aren't many attackers in the Premier League who can rival him, while his movement into the channels and aggression on the ball makes him so difficult to pin down.
While there are still some issues at times when it comes to decision making, that part of his game has improved too, and with better players surrounding him he would surely become more productive.
United have been struggling in front of goal lately, with main striker Romelu Lukaku enduring a torrid time, per Samuel Luckhurst of the Manchester Evening News:
Arnautovic would definitely offer something different to a United attack that has looked lifeless for long spells of the campaign. However, West Ham would surely be reluctant to sell their star man midway through the season; they're languishing down in 13th despite spending huge money in the summer.
Even so, given Arnautovic's temperament and age, United supporters would surely be hesitant about their team spending so much on him in January. The signing of Alexis Sanchez, who was the same age as Arnautovic when he joined United from Arsenal last January, has so far failed to pay off and would give United fans pause for thought on any deal for the Austrian.