And Spanish newspaper AS Diario has reported that Cheryshev has met with Rafa Benitez and club officials to see where he features in the clubs' plans.
The 24-year-old is yet to make a single appearance for the La Liga side since progressing from Real's B team in 2004.
And after loan spells at Sevilla and Villarreal, it's understood the Russian could be looking for the exit door if Benitez can't offer him a way into the first team.
Arsenal and Liverpool could both be possible destinations, with the Gunners needing more fire power and the Reds yet to replace Raheem Sterling.
However, the Premier League duo could have their work cut out to secure Cheryshev's signature, with Villarreal keen on another loan spell.
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