The midfielder has started only 11 games for the European champions this season, but replaced Luka Modric on the hour-mark at the Santiago Bernabeu after the Croatia international's participation was cut short due to injury.
Illarramendi has endured a frustrating time since making the move from Real Sociedad, but the 25-year-old is not looking to move on at the end of the season.
He said: "I have a contract here and I am very happy. I had a bit of difficulty adapting last year. However, now I am very happy here in the city and with my team-mates. Of course, I would like to stay here."
Gareth Bale also sustained an injury during a victory which ensured Real remain two points behind leaders Barcelona in the battle for the title.
Carlo Ancelotti's side face Atletico Madrid in the second leg of their UEFA Champions League quarter-final on Wednesday and Illarramendi is ready to take his chance if called upon.
He added: "They [Bale and Modric] didn't look good but we don't know what is wrong, we have to wait for the tests.
"They are two crucial players for us and we hope that there is nothing wrong with them.
"I am available and if they need me I will be there. I train as hard as I can every day for that chance."
