The Wales international has arguably been the Gunners' best player this campaign, contributing eight goals in 17 league appearances for the table-topping north Londoners.
Ramsey impressed for Arsenal once again at West Ham on Thursday, although his 23rd birthday ended with him limping off 65 minutes into his side's 3-1 win.
"It looks like a thigh strain," Wenger said. "For me, I don't know how serious the thigh strain is but the Christmas period is certainly over for him. It was his birthday.
"I don't know how long it will be. We will have to see [Friday] morning. He knew straight away, but we have to see."
Ramsey's injury somewhat overshadowed Arsenal's impressive comeback victory at West Ham.
Carlton Cole gave the hosts the lead against the run of play early in the second half, only for Arsenal to win through thanks to Theo Walcott's brace and a strike from the returning Lukas Podolski.
West Ham may have avoided that fate had influential defender James Collins not been forced off in the first half.
The centre-back had to be replaced due to a calf complaint and joins the lengthy injury list Sam Allardyce has to contend with heading into Saturday lunchtime's clash with West Brom.
"It is a calf problem, a tight calf," the Hammers boss said.
"He couldn't play on with it and I think that on top of all our other unfortunate injuries at the moment is of great concern to me.
"We just have to keep trying to get results between now and January, when we will try to bring in players of the right quality.
"We of course will also try to get the players fit as quick as we can.
"Ravel Morrison reported a groin [injury] on Christmas Eve and James we won't know how bad it is until [Friday].
"Then you have your Winston Reids, Stewart Downings, Richard Vaz Tes and Andy Carrolls that are progressing but are still a little way away at the moment."