The summer signing has stressed he must continue to improve under Carlo Ancelotti as he looks to secure a regular place in the starting XI at the Santiago Bernabeu
The 23-year-old has made 23 appearances in all competitions since his €39 million [£32.1m] summer switch from Real Sociedad, although he has struggled to fully justify his large price tag with his form to date.
However, the former Spain youth international is confident he will continue to improve under Ancelotti this season after he started the midweek win over Osasuna.
"I always want to give more, and I know I've got more to offer," the midfielder told reporters.
"I need to keep working hard and gaining confidence. I try to do my very best when I play."
Reflecting on the win at Osasuna, where he partnered the now fully-fit Xabi Alonso in midfield, Illarramendi was positive about Madrid's performance.
He said: "We're happy. We knew it would be tough if they nicked a goal, and luckily it was us who scored."
The 4-0 aggregate victory saw Real Madrid safely through to a quarter-final, where they will face Espanyol.