The 25-year-old has featured 144 times for the Gunners in all competitions since making his first-team debut in a 2-1 League Cup win over Manchester City in October 2004.
On Arsenal's official website, it said: "Everyone at Arsenal would like to wish Johan well for his time in Germany."
In his only two senior appearances in the current campaign, Djourou has captained Arsenal to Capital One Cup victories over Coventry City and Reading.
Hannover general manager Jord Schmadtke revealed on Friday that the centre-back was so determined to find regular playing time that he was not at all concerned over the precise terms of the contract.
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 Schmadtke.
"He fits the bill perfectly for what we were looking for because he is also eligible to play for us in the 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.