The 'Red Devils' remain on the lookout for a new defender having missed out on a host of targets during the summer and Milenkovic is understood to feature very highly in their thinking.
Indeed, United boss Jose Mourinho is believed to have personally watched the 6'5" stopper back in October when he lined up for Serbia against Montenegro.
The 21-year-old appeares to have done enough to keep himself in United's thinking, with another scouting mission planned for Sunday as they begin to plan their January business.
Milenkovic is contracted to 'La Viola' until 2022 and the Italian side are thought to value the youngster at around the £50m mark.
The man himself is understood to be aware of the interest and is close friends with United midfielder Nemanja Matic, with the pair international team-mates.
United are also considering other options, namely Leicester's Harry Maguire and Napoli's Kalidou Koulibaly, but it is thought unlikely that either of those sides will allow their star men to leave midway through the season.