The 25-year-old has become a bit-part player at the Bernabeu and the arrival of Mateo Kovacic from Inter Milan has led to the former Real Sociedad man not even making Rafa Benitez’s match-day squad for Real Madrid’s opening fixture of the season on Sunday night.
Marca state that Liverpool have made a €15m (£10.9m) offer for Illarremndi and that the Spaniard favours a move to Anfield over a potential return to his former club Real Sociedad.
Brendan Rodgers has enjoyed a busy summer in the transfer window and the further capture of a player of Illarramendi’s quality would offer Liverpool an added boost in their push for a top-four finish.
Liverpool won their opening two Premier League fixtures, albeit through narrow 1-0 wins over Stoke City and Bournemouth, and Rodgers could see Illarramendi become his eighth high-profile signing of the summer.
They managed a credible 0-0 draw away at Arsenal on Monday evening, but it remains to be seen whether they are yet credible contenders for a Champions League place.
Last term, Illarramendi made just ten La Liga starts, with former boss Carlo Ancelotti opting to use Luka Modric and Toni Kroos as his first-choice central midfield options. The arrival of Kovacic could prove to be the death knell for his Real Madrid career.
Rodgers has already brought in James Milner, Adam Bogdan, Danny Ings, Joe Gomez, Roberto Firmino, Nathaniel Clyne and Christian Benteke this summer, but has helped to balance the books with the big-money sale of Raheem Sterling to Man City.