The Switzerland international has played 144 times for Arsene Wenger's side since making his first-team debut in a 2-1 League Cup victory against Manchester City in October 2004.
The 25-year-old has made just two senior appearances this season, captaining the Gunners in their Capital One Cup victories over Coventry and Reading.
A statement on the official Arsenal website read: "Everyone at Arsenal would like to wish Johan well for his time in Germany."
Djourou's move to Germany is dependent on the formality of the necessary paperwork being completed as he has already passed a medical.
"Johan Djourou is an internationally-experienced footballer who we are convinced will be able to help us out quickly," said Hannover's director of sport Jorg Schmadtke.
"He fits the bill perfectly for what we were looking for because he is also eligible to play for us in the UEFA Europa League."
Coach Mirko Slomka has welcomed Djourou, saying: "After losing Felipe and Leon Andreasen (to injury), we needed to act defensively.
"With his physical robustness, his tactical understanding and his high level of commitment, I am convinced we will profit from having him."
Hannover are currently 11th in the Bundesliga.