The Russia international was linked with a move to Anfield for the majority of the summer, but a deal failed to come to fruition, leaving him at the Bernabeu.
And with the likes of Gareth Bale, Cristiano Ronaldo and James Rodriguez present in Madrid, the 24-year-old has struggled for game time, featuring on only two occasions in La Liga so far this season.
This has led to suggestions that Cheryshev is desperate to leave the club next month with game time vital as he plots to stay in his nation’s thoughts for EURO 2016, and Metro, citing Fichajes, claim that Liverpool are set to open talks with their continental counterparts soon.
Manager Jurgen Klopp is short of options on either flank, with Jordon Ibe the German’s only orthodox winger.
Although Cheryshev has failed to mark an impact at Real, he impressed during a season-long loan move to Villarreal last term.