Martinez limped off in the second half after attempting to shield the ball from Hammers defender Vladimir Coufal.
The Argentina international, 26, was playing in only his third league game after more than three months out with a foot injury.
"It doesn't look good," said Ten Hag.
"There is a bad concern. We have to wait for a few days for the diagnosis. He is very sad, very disappointed. We all feel really with him.
"First of all it is a personal disaster when it is really bad but also for the team it is really bad because he definitely brings us a lot."
Martinez missed 22 games in total with his foot injury before making his return against Tottenham on 14 January.
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