Ozil helped Real to victory in the Spanish Liga.
He told Bild: "It has been the best season of my career. The feeling of winning a title with Real Madrid was incredible, but the campaign hasn't finished yet. Now I want to win the European Championships.
"My feelings are good. Our goal is the title. If it wasn't we wouldn't go to compete. The conditions for preparing ourselves are perfect and we are totally concentrated on the tournament.
"We were runners-up in 2008 and third in 2010. We failed in our aim and, therefore, finishing second would be a disappointment."