The 26-year-old has played just 127 minutes of football this season due to the back problem and the thigh injury he sustained at Manchester United last month.
Manager Slaven Bilic said on Thursday that the striker was not close to a return.
"We have a few knocks but are expecting everyone to be fit except for Diafra Sakho who will be out for six to eight weeks," Bilic told reporters ahead of Monday's Premier League visit to second from bottom Swansea City.
"He will miss the Africa Cup of Nations and it is a big blow for us."
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Sakho also missed last year's tournament with a back injury.
Bilic re-iterated that 13th-placed West Ham do not intend to sell France midfielder Dimitri Payet who has been linked by media reports with a move in the January transfer window.
"We do not want to sell Dimitri Payet ... he is a great player. He has a contract and we will do everything to make sure he stays with us," the manager explained.
Bilic, though, said he hoped to sign a couple of players in the window.
(Reporting by Neil Robinson; Editing by Tony Jimenez)
