The Arsenal defender, 25, has fallen out of favour at the Emirates Stadium this season.
And boss Arsene Wenger gave the Switzerland ace permission to move to the Bundesliga.
Hannover are 11th in the table and will be hoping to push for a Europa League spot.
A Gunners statement said: “Everyone at Arsenal would like to wish Johan well for his time in Germany.”
The club added that the transfer was "subject to the completion of regulatory processes."