The Dutch international picked up the hairline fracture of the fibula in a tackle with Villa's Gavin McCann in the goalless draw at the Lane.
Teemu Tainio also suffered a sprained right ankle in the final seconds of the game and faces two to three weeks on the sidelines.
Jermaine Jenas, who had to be substituted just before half-time after suffering mild concussion in a clash of heads with Juan Pablo Angel, will be okay to resume training this week.
Martin told us this afternoon: "Edgar has a hairline fracture so he could be out next week but he's a tough guy so hopefully he'll be fit.
"Normally you would be out for a couple of weeks but with Edgar, he's tough and can play with a bit of pain. We'll assess him again later this week."Teemu got an injury late on and he's looking at two to two-and-a-half weeks but he's tough as well and wants to play.
"JJ has been doing well for us and had a big lump on his head, but he's fine.
"Edgar and JJ had to go off but I thought the players that came on, Michael Brown and Aaron Lennon, did very well."
Martin Jol reports that Edgar Davids suffered a hairline fracture against Villa on Saturday - but remains hopeful he could line-up against Fulham on January 31.