According to Calciomercato (h/t Lucas Sposito at Sport Witness), the Red Devils' interest could prompt Juventus into trying to agree a new contract with the winger.
Bernardeschi joined Juventus in 2017 from Fiorentina and signed a five-year deal with Massimiliano Allegri's side.
The Italian is a tricky, skilful winger, who can also play in a central role. He found it tough going forcing his way regularly into the starting XI in his first season as Juventus completed a league and cup double.
However, he's started the new campaign in strong form and has been one of Juventus' best players. He has two goals in four Serie A appearances and shone in the UEFA Champions League win over Valencia.
Juventus saw Cristiano Ronaldo controversially sent off at the Mestalla Stadium. However, they still prevailed 2-0 with Bernadeschi causing Valencia problems.
Football writer Adam Digby highlighted his contribution:
Allegri spoke about the youngster after the game and explained how he has developed during his time in Turin, per Sky Sport Italia (h/t Football Italia).
"Bernardeschi has become a real Juventus player and one worthy of the big games in a short period of time," he said. "He has grown so much in his maturity and technique, so I have no qualms now about playing him even in the biggest fixtures."
His performances have meant Paulo Dybala has been forced out of the team and onto the bench. Bleacher Report's Gianni Verschueren said he might have to get used to life as a substitute:
Manchester United possess a wealth of attacking options but do lack a natural right winger in the squad.
Football writer Liam Canning said they must strengthen that position in January:
Bernardeschi's performances will have caught the eye, but Manchester United may find it difficult to tempt him away from Juventus where he is starting to show exactly what he can bring to the Italian champions.