The midfielder acknowledges it will be difficult to secure regular playing time in his first season at the club, while Daniel Carvajal is optimistic ahead of the new campaign
The 23-year-old joined Madrid from Real Sociedad for a transfer fee of €32.19 million [£27.7m] earlier this summer, but he acknowledges he be far from a guaranteed place in the starting XI with the likes of Luka Modric, Xabi Alonso and Sami Khedira all vying for spots.
"I still have a long way to go to get into the starting team, which is very difficult in a team like this, but I'll try," the midfielder told reporters after making his Madrid debut in the 6-0 friendly win over Bournemouth.
"I concentrate on playing, I have very good people around me and I am very happy to play with them."
Illarra then went on to discuss his relationship with head coach Carlo Ancelotti and stressed he is pleased to be working with the former Paris Saint-Germain boss.
"He is very close with the players and has a good relationship with us. I like that sort of coach."
Fellow new arrival Daniel Carvajal, meanwhile, was equally positive about the Italian boss and is confident that Madrid's policy of signing young Spanish players will pay off.
"The coach is quiet and gives us conviction and confidence. This first week has been great," the full-back said.
"We are a young team with a low average age. Spanish players have been brought in this year and I’m sure it will work out well."