Kovacic has spent the last three years at the Bernabeu, making 109 appearances in all competitions, but half of the Croatian's outings have come from the substitutes' bench.
The former Inter Milan player has been linked with a number of different clubs around Europe, and the 24-year-old says that it is time for him to gain regular first-team action elsewhere.
Kovacic told Marca: "I'd like to play more, I love football and being on the pitch. I know that playing as a starter in Real Madrid is not easy, especially when you're very young.
"I get it, but I also think that the best thing to do is to go to another club where I can get more playing time. I've been at Real for three years, I haven't played much. I'm not happy about it, that's why I think it's time I leave.
"Real Madrid is an amazing club, with many great players and maybe I'm too young to give a regular contribution to the team"
Kovacic still has three years remaining on his contract with the European champions.