The Spanish midfielder is on the fringes at the Santiago Bernabeu and is reportedly interested in a move to Anfield.
The former Real Sociedad playmaker joined the capital city side and was initially envisaged as a long-term replacement for Xabi Alonso due to their similar attributes and playing styles.
However, this has not quite materialised for Illarramendi, who has found first-team football difficult to come by this season.
The 25-year-old has found himself behind Luka Modric, Isco and Toni Kroos in Carlo Ancelotti’s thinking this season, with Sami Khedira and Lucas Silva also in the reckoning for a starting XI place.
With a desire to play more first-team football next season, the Spanish newspaper have stated that the midfielder has taken a liking to the idea of moving to the Premier League.
Although Arsenal are reportedly also interested according to Marca, Illarramendi would supposedly prefer a switch to Anfield.
This is regardless of the likelihood that Brendan Rodgers’ men will not be in the Champions League next season, with the Reds’ longstanding link to Spanish players having a sizeable role in his decision.
With Steven Gerrard leaving Merseyside this summer, Rodgers is expected to try and replace him in the transfer market ahead of next season.