David Ospina suffered a knock in Sunday's draw against top-four rivals Manchester City.
And, according to the Mirror, he is a major doubt to face the Hammers.
Arsenal boss Wenger is already without first-choice Petr Cech, who is set to miss the next two weeks with a calf injury.
So No.3 keeper Emiliano Martinez is in line to start at the Emirates.
The 24-year-old has played just three times this season, with all his appearances coming in the EFL Cup.
Despite his lack of game time, Wenger believes Martinez can be Cech's long-term successor.
“If you look at him, I personally see all the top qualities that a goalkeeper needs,” Wenger said in November.
“He has the stature, he has the passion, he has the technical quality, he has the decision-making, he has huge power and he’s not frightened of anything."
