The Gunners have wanted to sign the former Borussia Dortmund star on loan and stepped up their interest after Monday’s sale of Alex Song to Barcelona.
An agreement has now been reached between Arsenal and Real and the move could happen before the weekend if Sahin wants to make the switch to Emirates Stadium.
Liverpool had also been linked with 23-year-old Turkey international but Arsene Wenger’s Gunners have always been favourites.
Classy midfielder Sahin has struggled at Real since arriving in 2011 but could be considered the final piece of Arsenal’s summer redesign, which has seen significant changes in personnel in London.
Former captain Robin van Persie was sold to Manchester United, while Song has joined Real’s Spanish rivals, Barcelona, among the reshuffle.
But Wenger, who came in for criticism in Saturday’s season-opening 0-0 draw with Sunderland, has also brought in Lukas Podolski, Olivier Giroud and Santi Cazorla.