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."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.