The 26-year-old had spent his entire career with the former Spanish champions, but made just seven league appearances for the first team.
"After 16 years I have put an end to my time in this, my second home - Real Madrid. This has been a time and place where I have grown, as a person and a player and in colours that I have given my best to," he said in a statement.
"At 26, I have a long road ahead where I believe that the best is yet to come. A few have walked beside me for a significant part of my life, my sincere thanks for your dedication and support, my teammates until a few hours ago, I wish you every success. To the whole Madrid family, a hug, friends."
As a free agent, Adan is still eligible to join a new club after the transfer deadline.