The 29-year-old, who suffered the injury in Friday's 1-1 Bundesliga draw with Hertha Berlin, had an operation in Basle on Monday, his club announced.
Hannover added in a statement on their website: "Ya Konan will return to Hannover shortly after appropriate aftercare on site.
"Following the operation and the extensive rehabilitation, he will be out for a long time and not play again in the Bundesliga before the winter break."
The loss of the Ivory Coast international is a blow to Hannover, have had an impressive start to the season. They lie fifth in the table after eight games.