The Brazil international joined the Bianconeri from Porto in 2015 and has won the Serie A and Coppa Italia double in each of his full seasons in Italy thus far.
Shaw, meanwhile, has made just 16 appearances in all competitions this term for the Red Devils and has found himself at loggerheads with manager Jose Mourinho on numerous occasions.
According to the Manchester Evening News, Matteo Darmian's possible move to Juve could enable United to negotiate a player-plus-cash deal for Sandro, whose contract expires in 2020.
The report claims that Mourinho is unsatisfied with Shaw's progress this season and, alongside Darmian and Daley Blind, is set to exit this summer as part of the Portuguese's defensive overhaul.
Shaw has made 63 appearances for United since his 2014 move from Southampton.