The Portugal international, 31, comes off contract at the end of this season, so leaving Juve to discuss pre-contract terms with Bernardo's management team from January 1.
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Tuttosport says Juve are making their move and contact has already been established with Bernardo's camp.
The veteran midfielder has come close to leaving City over the past two summers, only to be convinced to stay by manager Pep Guardiola.
In Turin, it's claimed City are unlikely to offer new terms to Bernardo, so leaving him free to choose his next club from the beginning of the New Year.