BILD says Arsenal will announce the signing of Leno on Tuesday.
The 26-year-old will be presented as a Gunners player having agreed terms on a long-term contract.
Leno is seen by new Gunners boss Unai Emery as the perfect addition to his squad following Petr Cech's uncertain performances in the latter months of Arsene Wenger's reign, with David Ospina also set to be allowed to leave.
Germany international Leno has six caps for his country, but isn't with them at the World Cup.
Leno did have Gunners' fans excited last week when he "liked" an Instagram post linking him with a move to the Gunners.