The Colombian has less than 11 months remaining on his contract with Los Blancos and the club are likely to sanction his sale rather than risk losing him on a free transfer next year.
Galatasaray have recently been credited with an interest in the former AS Monaco star, but it is reported that the Toffees are still leading the race to sign the 29-year-old.
Rodriguez has previously played under Toffees boss Carlo Ancelotti at Real Madrid and Bayern Munich and he is said to be keen on a reunion with the legendary Italian manager.
Ancelotti had aimed to sign him when he was the head coach of Napoli last summer, but manager Zinedine Zidane decided against permitting his departure.
However, with the current contract situation, Rodriguez is bound to head for the exit door and the Merseyside outfit are in an advantageous position to pursue his services.
Rodriguez had a difficult 2019/2020 season with the Spanish champions and he was restricted to only 14 appearances across all competitions.
Should he make the move to the Toffees, he should improve their attack with his creativity, particularly from his preferred number 10 position.
The only concern will remain whether he can stay unscathed. Over the past couple of years, he has suffered as many as nine short-term injuries.
Rodriguez, who is valued at £28.8m, as per Transfermarkt, made just one appearance for Los Blancos after their 2019/2020 season resumed in June.
He personally requested Zidane to leave him out of the matchday squad for the final seven games of the campaign after falling out-of-favour.
Stats from Transfermarkt.com