According to Calciomercato (h/t Sport), the Costa Rica international is disappointed at losing his place in the team to Thibaut Courtois and believes he does not have the trust of coach Santiago Solari.
The 31-year-old is interested in a move to Turin, which would see him follow in the footsteps of former team-mate Cristiano Ronaldo, who joined the club from Real Madrid in July.
Navas joined Real Madrid from Levante in 2014 and has gone on to win La Liga and three UEFA Champions League titles with Los Blancos.
Despite his success at Real Madrid, Navas has never been widely recognised as one of the world's elite goalkeepers.
It was not a huge surprise when Real Madrid opted to bring in Courtois from Chelsea over the summer for a fee of £35 million, per BBC Sport.
Former manager Julen Lopetegui had rotated between Navas and Courtois. However, new boss Solari appears to have installed Courtois as his first choice:
Sports journalist Robbie Dunne noted Courtois' form at Real Madrid:
Navas now looks set for a season on the bench at Real Madrid. He did play in the Copa del Rey first-leg win over Melilla, but that seems to be the only competition he will feature in unless Courtois picks up an injury.
The Costa Rican may be unwilling to play second fiddle at Real Madrid for long. He will turn 32 in December and is heading towards the end of his career.
It remains to be seen if a move to Juventus would yield any more game time for Navas. The Italian champions have Wojciech Szczesny as first choice and Mattia Perin as back-up.
Szczesny has enjoyed success in Italy since leaving Arsenal for a loan spell with Roma. He kept Alisson Becker on the bench during his time in the capital, before moving to Juventus in 2017.
However, he has come under scrutiny for his performance in the 2-1 defeat to Manchester United. The Poland international failed to deal with Ashley Young's free-kick into the penalty area, which led to United's winner and drew criticism:
It was not Szczesny's finest moment at Juventus, but he has been dependable during his time in Italy.
Football commentator Matteo Bonetti noted how Perin is also a strong option for Juve:
Matteo Bonetti @TheCalcioGuy
@sidlowe @razahyder1947 @MiguelDelaney Anyway, if Szczesny were to have a serious dip in form, Juve have an excellent backup in Mattia Perin who joined over the summer wanting to be Juve’s number 1. Has the quickest reflexes in the league and makes a lot of Hollywood type saves. Only problem is he’s rubbish with feet
Juventus have also been linked with a move for Manchester United's David De Gea, per The Times (h/t Football Italia). De Gea is one of Europe's best goalkeepers, and a move for the Spaniard would make more sense if Juve wanted an upgrade.