The Spanish midfielder, who surprisingly, is still yet to win his first senior cap, has struggled to retain a foothold in the Real Madrid line-up since his move from Real Sociedad two years ago.
Last season, Illarramendi started just ten La Liga fixtures with former boss Carlo Ancelotti opting to use Toni Kroos and Luka Modric as his regular central midfield options, and newly-appointed manager Rafa Benitez’s decision to bring in Kovacic will further restrict any involvement for the 25-year-old battler.
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El Mundo Deportivo claim Real Madrid will look to recoup as much of the €40m (£28m) they paid for Illarramendi in July 2013.
Brendan Rodgers has invested well over the summer, and the sale of Raheem Sterling to Manchester City has largely helped to cover the outlay to bring in a host of top talents.
The Anfield side brought in free agents James Milner, Adam Bogdan and Danny Ings and spent well to sign young defender Joe Gomez and big-money additions Christian Benteke, Roberto Firmino and Nathaniel Clyne.
Rodgers saw his side fall from their runners-up spot in 2013/14 to a sixth-placed finish last season, and the Northern Irish manager will hope his side can once again challenge for a top-four spot during the current campaign.
They are one of only four sides to enjoy a 100 per cent success rate in the division so far, and Illarremendi’s arrival might just be the boost the Reds need to mount a legitimate title challenge.