The Dutch centre-back has struggled in Turin since arriving from Ajax last summer in a deal worth £67.5m.
United, along with Barcelona, were the clubs attempting to sign De Ligt before he decided to join the Old Lady.
According to the London Evening Standard, United have been watching the 20-year-old's progress all season.
Any deal for De Ligt could be made easier by Juve's interest in Paul Pogba.
The World Cup winner is reportedly angling for an exit from United, with Real Madrid also interested in his services.