The Daily Telegraph is reporting that the Red Devils are keeping an eye on the 29-year-old striker as a potential target and that they play on keeping tabs on him until the end of the season.
However, the same story says that the Red Devils are “far from certain” about making a move to sign the Austrian next year and that they are simply monitoring his form as things stand.
According to the same story, Manchester United’s interest in Arnautovic may depend on whether Jose Mourinho remains as the Red Devils boss beyond the summer.
Arnautovic has scored five goals and made one assist in 11 Premier League games so far this season.
The Red Devils boss has found himself under pressure in recent weeks following his side’s stuttering start to the new season.
Manchester United dropped yet more points in the Premier League on Saturday when they were held to a goalless draw by Crystal Palace at Old Trafford.
That result left Manchester United in seventh place in the Premier League table and 14 points off leaders Manchester City.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news