The Spanish outfit finished second in La Liga last season, while they were also frustrated in their attempts for Copa del Rey and Champions League success.
However, the German is confident that the capital outfit will enjoy success under new boss Carlo Ancelotti next season.
"We have big goals at Real Madrid. And we can only achieve that with great players. We're aiming for first place in all three competitions. This is not about arrogance," Ozil told Sport Bild.
"I look forward to the new season. At Real Madrid you cannot compete for a whole season with just 11 players. We need the best quality.
"Until now, I have always succeeded. In three years at Madrid - and also previously in the Bundesliga - I have played more than 100 matches. This is almost as good as it gets."
Ozil joined Madrid in 2010.