Jurgen Klopp is a long-time admirer of Rabiot and tried to sign him when he was Borussia Dortmund coach.
The combative midfielder has found himself starting on the bench for Juve's opening three games of the season.
The 23-year-old is in the final year of his contract in Paris and has, for the moment, rejected all the offers of a new deal from the club who are still hoping to convince Rabiot to pen a new contract.
The website reports that the Reds have been in contact with PSG midfielder Adrien Rabiot's agent, who happens to be his mum, over a potential free transfer in the summer and that Klopp and the player have spoken.
A France global, Rabiot is considered a key target by Barcelona, who see him featuring heavily in their plans.
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Osaka will next face Lesia Tsurenko, the 36th-ranked Ukrainian who beat Czech teenager Marketa Vondrousova 6-7 (3-7), 7-5, 6-2. On Monday, Roger Federer and Maria Sharapova crashed out in back-to-back seismic shocks under the Arthur Ashe Stadium lights.
The Anfield atmosphere clearly had a profound impact on Rabiot that night, and he would have been impressed with what he saw on the pitch too as Liverpool beat City 3-0 on their way to a 5-1 aggregate triumph.
Former Liverpool and England left-back, Stephen Warnock has identified the club's defender, Andy Robertson as one of the most influential players in the Premier League this season.
Clubs have been able to sign players without work permits from within the European Economic Area (EEA).
Rabiot has been linked to Tottenham ever since he was a teenager and he has grown as a player considerably since, forcing his way into the France global team.
That would be another excellent addition to the Reds squad but they'll no doubt face competition from Barcelona and other clubs.