Reds and Gunners go head-to-head for skillful attacker.
The Russian international has been a top performer in La Liga in recent times, though he has only managed to truly showcase his ability on loan at Villarreal last season.
Cheryshev faces plenty of competition to get into the Real Madrid team on a regular basis, and this could mean he tried to leave the club for a move elsewhere in January.
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According to the Daily Star, Liverpool seem intent on bringing him to Anfield for a fee of around £15m, though Arsenal and Everton have also shown a strong interest in the past.