The promising midfielder is being linked with a move to los Blancos, but the Basque club's captain says they will have to be offered a lucrative deal to lure him away
The Basque outfit will not let go of their prized asset unless the €30 million (£25.8m) release clause in his contract is triggered.
Illarramendi was one of the stars of this summer's European Under-21 championship as Spain were crowned champions and new Madrid coach Carlo Ancelotti is supposedly a keen admirer of the holding midfielder.
"We don’t know what Asier's intentions will be, but if Madrid intend to pay his release clause it’ll be very difficult to keep him," Xabi Prieto told AS.
"At the moment, we don’t know anything, but the club have already made it clear that if Madrid want to buy Illarramendi, they’ll have to pay his release clause. And it seems that Real Madrid are willing to pay it.
"For the moment, he is a Real Sociedad player, though. We’re talking about a hypothetical situation. We’ll respect his decision. With or without him, the team will still be strong."
Illarramendi's contract with Sociedad runs until the summer of 2018.