Sahin reveals that he chose Real Madrid because they are the greatest club in the world
The 22-year-old announced his decision to part company with BVB on Monday morning. Madrid later confirmed they had reached an agreement via their official website.
The German-born Turkish international hopes to succeed at the Bernabeu and understands it will not be easy.
"Money wasn't the reason for my move. When I'd stay healthy, I'd earn enough anyway," he told Dortmund's official website.
"Real Madrid is the biggest club on earth. I'm proud to make this step. It's the highest plateau you can reach as a football player and the chance for the world career I've hoped for. Only because of this I will leave Dortmund."
Despite his eagerness to impress in Spain, the Bundesliga club will not be forgotten any time soon.
"Not many clubs were an option to give this up here. The team is in my heart. The last three years were great in this squad, where everybody gave everything for each other.
"I'm a player who has the potential to break through at Real Madrid. I've signed there for six years. It was hard decision, because I'm a Borussian and love Dortmund and the club - and I always will be a Borussian.
"It made me very proud, that I was allowed to lead this team as captain into this stadium. I want to say thank you to all, who supported me on my way.
"The hardest part for me was to announce my decision. I will miss Jurgen Klopp. I've invited the team, trainer, staff members, plus Mr Watze and Mr Rauball to the next Clasico."
Real Madrid will reportedly pay the German club around €10 million for the transfer.