The former - who has been a regular fixture since re-joining from Milan in the close-season - is unlikely to play a part having sustained a groin injury against Crystal Palace on Saturday.
Flamini was forced off the field after just eight minutes in Arsenal's 2-0 win, with Wenger now facing an anxious wait to determine his availability.
"He has a scan today and then we will get the results as to how long he will be out," he said.
"It doesn't look like a bad muscle injury but honestly it's too early to give you a precise answer. It's a groin injury."
Wenger could also have to make do without teenager Gnabry - who signed a new long-term contract on Monday - can play a part in the fourth-round encounter.
The German replaced Flamini at Selhurst Park, but also picked up a knock.
"Serge Gnabry has a problem, he got a kick on the ankle. He will be assessed tomorrow (on Tuesday) ," Wenger added.
"After that we will decide who plays and who doesn't play."
Wenger confirmed that Jack Wilshere is fit and available, but the England midfielder could be rested, with the Arsenal boss also revealing he is set to pick a rotated side for the game at the Emirates Stadium.
