Playmaker Sahin, 22, helped steer Dortmund to this season's Bundesliga title and will join the La Liga giants as they seek to end Barcelona's grip on the domestic title and chase a 10th Champions League triumph.
While the clubs did not disclose how much Real agreed to pay, his contract with Dortmund included a clause that allowed him to leave if there was an offer of 10 million euros or more from outside Germany.
Sahin, who has missed his team's last three matches through injury and is expected to be out for six weeks, said he was very proud to make the move to the Spanish capital.
"The decision was hard to take. I am Borussia and will remain Borussia," he told reporters.
"I will miss the Bundesliga and my homeland. I would like to thank everyone who has accompanied me on this journey until now," added Sahin, who has been at the club since the age of 12.
Sahin's arrival and the possible purchase of Altintop could mean that Real midfielders Fernando Gago, Sergio Canales and Pedro Leon, little used this season, will move on.
