Illarramendi has struggled to nail down a first-team place since leaving Sociedad for the Bernabeu in the summer of 2013 and is expected to move on this summer.
Madrid were keen to cash on in him in January, but Athletic Bilbao were the only club willing to make a firm offer and Illarramendi was not keen on the idea.
Liverpool are reported to be monitoring the player but Moyes has hinted that Sociedad could offer Illarramendi a return to his former club.
Speaking at a pre-match press conference, Moyes said: ???We want to get good players here.
“Illarramendi is a very good player and the president, myself and [sporting director] Loren, we want to improve the standard, we want to get good players here, we want more competition for the team.
???We have to make sure we can compete with the top six or seven teams in Spain. We want it quickly. We???re going to have to make sure we add to the squad and improve it with good players. And the player you mentioned (Illarramendi) is certainly a good player.???