He returned to first-team action in the loss to Tottenham Hotspur on Nov. 22 after more than a month out with a thigh injury.
"We are still waiting for the results of his (Sakho's) scan, but it will definitely not be a couple of weeks -- four to eight, maybe longer," Bilic said at his pre-match news conference on Thursday.
"We are waiting for the opinion of a couple of specialists. He has a bit of history with that thigh. It's not a light one unfortunately."
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"Of course it's a big blow. We are not bad in terms of quantity of injuries, but unfortunately they have happened to key players," Bilic said.
Sakho joins midfielder Dimitri Payet and Enner Valencia on the treatment table.
Valencia (ankle), has not played since Nov. 7, but is expected to return before the end of the year, while Payet (ankle) is out until mid-February.
"Enner is having a good recovery so hopefully we will have him in a couple of weeks time. It's a good chance for the guys to step in," Bilic said.
The Croatian also hinted that he might hand striker Andy Carroll a rare start when the eighth-placed Hammers travel to Old Trafford to take on third-placed Manchester United on Saturday.
"Andy has lifted our game when he's come in for the last 20 or 30 minutes and now he needs to show that from the start," Bilic said.
(Reporting by Simon Jennings in Bengaluru; editing by Sudipto Ganguly)
