The Manchester Evening News state that The Red Devils are exploring the possibility of bringing in the Italian defender to add to their defensive ranks at Old Trafford.
Whilst the Paul Pogba saga continues to rumble on, Mourinho is keeping his eye firmly set on The Old Lady and a double swoop on Turin.
Bonucci starred at Euro 2016
The 29 year-old was a constant fixture in Antonio Conte's starting XI in France and conceded just twice on their way to a penalty shoot-out elimination by Germany.
A product of Nerazzurri youth academy with Inter Milan, Bonucci has been with Juve since 2010. During that time the central defender has amassed 198 appearances, just two short of 200 competitive games for the five-season Serie A champions.
Mourinho looking for further experience
United have already recruited defensively this summer, with Eric Bailly the first addition of the Mourinho era in Manchester.
Despite the £30m spent on the Ivorian defender from Villarreal, the Portuguese has already suggested that Bailly may not be ready to Premier League football straight away.
With the future of Argentine Marcos Rojo lying elsewhere and with makeshift defender Daley Blind also looking surplus to requirements, the addition of further numbers in the back line may not raise too many an eyebrow.
United will face strong competition from their cross-city rivals Manchester City for Bonucci, as Pep Guardiola looks to put his own stamp on the city, with moves for Everton's John Stones and Athletic Bilbao's Aymeric Laporte thus far finding no resolution.