The 25-year-old has failed to hold down a regular spot in the Madrid side and has fallen even further down the pecking order following the addition of Matteo Kovacic.
Liverpool have been long-term admirers of Illarramendi, however Marca are reporting the Merseyside club face competition from David Moyes for the midfielder's signature.
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The Spanish publication have claimed Sociedad are preparing to match Liverpool's €15 million bid for the Madrid player, but claim Brendan Rodgers' side are still favourites for his signature and are sitting pretty in the knowledge the Spanish giants have to sell.
The Spaniard would be given a heroes welcome if he was to return to Sociedad, a club where he broke through the academy and went on to play 56 times for the first-team. However he is more interested in a move to England and the possibility of testing himself in the Premier League.
There has been talk that Lucas Leiva is on his way out of Anfield having fallen down the pecking order at Merseyside, however if the Brazilian was to leave a replacement would have to ushered in.
Rodgers has opted for a midfield partnership of Jordan Henderson and James Milner in the opening weeks, but with the Liverpool captain out injured, Emre Can is expected to come in with Lucas being handed a spot on the bench.
This signifies Liverpool's shortcomings in the middle of the park and will need to bring in a replacement for Lucas, with a move away still looking increasingly likely.
Illarramendi would be the perfect replacement as he brings a wealth of experience from the top level and is still at an age to improve.