Wilkinson was forced off the field near the end of England's 16-12 victory over Scotland - with the fly-half having subsequently had a scan on his right forearm.
Unfortunately the results were inconclusive, the England medics needing another 24 hours at least to determine the extent of the injury.
Johnson said: "Jonny got scanned last night on his upper forearm, elbow region.
"That didn't show anything and while that is good news, obviously at this point in the tournament it is a little bit of a worry.
"We need another day to see where he is and whether he will be available for the weekend."
Centre Mike Tindall is also a doubt for the clash with Les Bleus after he suffered a dead leg against Scotland.
England will wait another 24 hours before deciding when he can return to training.
England will hope the news about Wilkinson is not as serious as that which has hit New Zealand, with their star fly-half Dan Carter ruled out of the tournament with a groin injury.
"With the news today (about Carter) there is concern for us with injuries," Johnson said.
"Generally we are in a pretty good place. We will hope for the best in the next 24 hours."
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