Madrid won a Champions League and Copa del Rey double last season, but fell short of capital rivals Atletico Madrid in La Liga.
"We all want more and of course, I do too. The balance of the season for the team has been very positive," he told AS.
"Can we do better than what has been achieved this year? We have won two trophies, including La Decima, which we'll remember for a long time. I am very happy about that.
"But improving what the team has achieved this year will be very difficult."
Illarramendi joined Madrid from Real Sociedad last summer.