The Cote D'Ivoire international, who arrived at the club in 2014, has reportedly turned down PSG’s offer for a three-year extension.
United are looking to recruit competition for Antonio Valencia at right-back while Bianconeri are in need of a new right-back after Dani Alves revealed he would be terminating his contract to join Manchester City.
The report suggested that Juve have turned their attention to the 24-year-old who made 22 Ligue 1 appearances last season.
Aurier is not on good terms with PSG coach Unai Emery and is keen to leave the side having made 161 appearances in the French top tier and scored ten goals.
Sporting director Antero Henrique has already said that he doesn’t keep players against their will and it remains to be seen if he will sell the Ouragahio-born to the Premier League side or the Serie A champions.
The former Toulouse defender currently has a year remaining on his contract with Les Parisiens.