The 23-year-old has three caps for Côte d'Ivoire and is on loan to Hanover from Bayer Leverkusen, but underwent surgery on Thursday to repair his right knee which he injured in training.
"Only with the ideal healing process would Constant play for us again this season," said Hanover coach Mirko Slomka. "The chances are minimal."
Djakpa has played regularly for Hanover, who are 16th in the Bundesliga, and made 23 appearances in the German league this season. Côte d'Ivoire have drawn Brazil, North Korea and Portugal in Group G at the FIFA World Cup.