The Toffees were said to be keen on the player this summer.
It is reported by The Daily Star that Cheryshev, who has a £147 million buy-out clause in his current Real deal, became an option for Everton after it became clear their pursuit of Andriy Yarmolenko was going to prove futile.
Cheryshev impressed during a loan spell with Villarreal last season and with competition fierce at Real Madrid he has been consistently linked with a move away from the Bernabeu this summer.
Instead he has stuck around, with new Real boss Rafa Benitez believed to be a big fan of the Russian international – however, he is still waiting for his La Liga debut with Los Blancos.
In the end Everton went on to sign their former loanee Aaron Lennon from Tottenham on a permanent deal, but in an ideal world Everton would love to have had both at their disposal ahead of a hopeful push back towards Europe.
Cheryshev must now hope he can force his way into the Real Madrid reckoning