Illarramendi, who was confirmed as a Real player on Friday afternoon, said that he is now looking forward to challenging for silverware with his new club.
"The decision was not easy, but it was a unique and irresistible opportunity," Marca quotes Illarramendi as saying.
"At first I was flattered by Real's interest. They are interested in you and excites you. Real is a big club, which fights for titles."
Illarramendi recently won the European Under-21 Championship with Spain.