The 23-year-old, who has also been targeted by Liverpool this summer, is set to join on a season-long basis with a view to a permanent move.
His arrival should be confirmed early next week as Arsenal look to replace Alex Song, who is on the verge of a £15m transfer to Barcelona.
Turkey international Sahin made 11 appearances for Real last season.
He joined the Spanish giants from Borussia Dortmund on a six-year contract in May 2011 but has struggled to establish himself at the Bernabeu.
Liverpool manager Brendan Rodgers, who worked under Real coach Jose Mourinho at Chelsea, was hopeful of pushing through a deal for Sahin last week.
Real Madrid’s former Liverpool midfielder Xabi Alonso has also attempted to persuade Sahin to move to Anfield.
But Sahin is set to join Arsenal, who started the season with a goalless draw against Sunderland on Saturday.