The 25-year-old was lauded for his performances last season as Los Blancos won the league and Champions League double, but his displays this term have led to questions over his long-term future at the club.
Madrid's struggles so far in 2017-18 will reportedly lead president Florentino Perez to conduct a major overhaul of his squad when the transfer window reopens, with the likes of Cristiano Ronaldo, Gareth Bale and Karim Benzema among those to have been linked with an exit.
Isco has also been touted as a possible victim of the clearout, with the Spain international having previously been linked with a move to the Premier League should he leave the Bernabeu.
Teamtalk now claims that Liverpool have emerged as the latest contenders for his signature, with Jurgen Klopp identifying the midfielder as an ideal replacement for Philippe Coutinho, who joined Barcelona in January.
However, Manchester United are also understood to be monitoring the situation of a player who has scored five goals and set up six more in 30 appearances across all competitions this season.