The Spaniard joined Los Blancos from fellow La Liga outfit Real Sociedad back in 2013 after his buyout clause was activated, but the 25-year-old has found it difficult at the Bernabeu, struggling to push ahead of the likes of Luka Modric and Isco in the midfield pecking order.
The Reds are believed to be willing to give Illarramendi the chance to revive his career at Anfield, and have a lodged a £15.7 million bid for his services.
With Real adjudging the player to be surplus to requirements at the club, and with Illarramendi himself keen to move on, a deal could be struck sooner rather than later.
Liverpool have been one of the busiest Premier League teams in the transfer window so far, recruiting no fewer than four players. They are expected to make many more additions before the new campaign as they seek to avoid a repeat of last season.