The report claims the Italian champions are in negotiations with United over a deal for the Portugal international, and Vucinic is on the table.
The Montenegrin international has fallen out of favour at the Juventus Stadium following the acquisitions of Carlos Tevez and Fernando Llorente.
He's made just three starts for the defending Serie A champions in the league this season, and has been linked with a January exit.
Nani, who only signed a new deal at United over the summer, is under contract at Old Trafford until the summer of 2018.
But that hasn't put Juventus off, with the club confident they can convince United to sell.
The Portuguese winger was booed off the pitch upon substitution against Stoke City earlier this term, and he's become a lightning rod for criticism of the club's inconsistent play this term.
The report fails to elaborate on the Vucinic for Nani offer, but it's unclear whether Moyes would even consider the deal.
The defending Premier League champions have plenty of options up front, including Javier Hernandez, Wayne Rooney and Robin Van Persie.