The Spanish midfielder is out-of-favour at the Santiago Bernabeu, with the Scottish boss stressing he is welcome back at Anoeta.
The Spanish midfielder has fallen down Carlo Ancelotti’s pecking order at Real Madrid and is victim of Isco’s excellent form at the Santiago Bernabeu.
Illarramendi has only started three La Liga games this season and as such will surely be frustrated with his lack of opportunities to play.
Although the Gunners are being accredited with an interest in the star, former club Real Sociedad could well be suitors also as David Moyes has admitted that he would welcome the player back to Anoeta.
"I know the player and his qualities well,” Moyes stated in a press conference, quoted by AS.
“When I was at Everton I looked at him. It seems that Real Madrid don't want to use him and I'd be delighted if he came here, but we need all the information from Madrid so that we can see what might happen."
Illarramendi was a product of the Basque outfit’s youth academy and made 54 La Liga appearances for Sociedad before moving to the Spanish capital in a deal worth a reported €32 million in 2013.
The 24-year-old has not been capped by Spain’s senior squad as yet but played a key role in his nation’s under-21 side that claimed victory at the European Championships in 2013.