"He is a great player as is everyone that I will play with. Xabi recommended that I come, he told me it was incredible and so it is proving," said Illarramendi.
"He already told me that this was impressive and I am seeing that it is all true. I have the opportunity to now play with him.
"It is clear that Xabi Alonso is a great player, he came through the youth team at La Real and has always been a reference point. I hope to learn a lot from playing with him."
Illarramendi becomes Madrid's second signing of the summer following the arrival of Isco from Malaga earlier on in the transfer window.