The German club announced via Twitter that the transfer has been agreed, which makes the goalkeeper Unai Emery's second signing as manager.
The arrival of the German international, who missed out on a spot in the World Cup squad, calls into question Petr Cech's position at the Emirates.
The former Chelsea stopper was recently handed the number one jersey, but Leno will aim to be first choice at the North London outfit.
German publication Kicker recently reported that the fee agreed was around £21.9m, but the figure has not yet been revealed by either club.
Leno spent seven years at Leverkusen, but did not win a trophy during that entire time.