The Argentina defender will play on-loan at Racing Santander, though has been mentioned as a potential replacement for crocked Real defender Fabio Cannavaro.
Garay said: "There was the possibility to replace Cannavaro after his injury. It's not the best way, but I cannot deny that I would have loved the chance.
"I would be much happier if I had gone to Madrid directly, but I also acknowledge that I will be very proud to play with Racing in Europe."
He also said: "I joined Real Madrid for two reasons: because Madrid was the most interested in me and really pushed for my signing and also because this is the best team in the world.
"The other clubs were no more than rumors. And you know: you can never say no to Madrid."