The Spaniard, who has emerged as one of West Ham's most consistent performers this term, was injured against Leicester on Saturday.
Obiang, 24, appeared to roll his ankle during the second half of the 3-2 defeat at the London Stadium and was carried off on a stretcher.
West Ham had already lost defender Winston Reid to an adductor injury in the first half, and he is due to undergo a scan later this week.
However, the club fear their New Zealand international could face around six weeks on the sidelines.
In a triple injury blow for manager Slaven Bilic, winger Michail Antonio suffered a hamstring problem which has forced him to withdraw from the England squad to face Germany and Lithuania.
Jack Sullivan, the son of joint-owner David Sullivan, tweeted: "Obiang is out for the rest of the season
"Antonio 3-4 weeks Hamstring, but as he's a superb athlete we hope it might be sooner.
"Reid scan later this week, but at least 4-6 weeks and might be longer."
West Ham travel to Hull after the international break and hope to welcome back captain Mark Noble, who missed the Leicester game with a dead leg.