The injury to Wilkinson is certain to reignite the club versus country row, with Ashton convinced England's international stars should be rested in between Six Nations matches.
"He's just got cramp and that's why we took him off," said Newcastle director of rugby John Fletcher.
"Whether the cramp was anything else we will have to see, but he's icing it as obviously you would. It's too early to say, he's got cramp in his hamstring, so we took him off.
"Basically all you can do with that sort of acute injury is ice it, there's nothing else you can do for 24 to 48 hours.
"You're going to have to see where the injury is, so he'll be in our care. I'll talk to Brian Ashton and Rob Andrew and no doubt they'll be in the changing room right now.
"The problem with Jonny is that he has never pulled a muscle, he's done one or two other things but he's never pulled a muscle, so he does not have an understanding of the hamstring area.
"I would imagine that there are some damaged fibres so we will just have to wait and see how bad it is, it'll just be a time thing."
While Ashton may be without the mercurial boot of Wilkinson against the French, London Irish's Shane Geraghty proved he is ready to replace the injury-plagued Newcastle star after racking up 23 points at the Madejski Stadium.
The 20-year-old was included in Ashton's England training squad last month, before dropping down into the Saxons after Wilkinson's shock return from injury.
And despite the return of England internationals Toby Flood, Mathew Tait and Wilkinson to the Falcons' back line - Geraghty masterminded a fourth domestic win on the spin for Irish, scoring two second-half tries for the Exiles in the process.
"Obviously Shane Geraghty has put his hand up at the right time with Wilkinson having a bit of an injury," said London Irish chief Brian Smith.
"He's one of the best in the country and maybe he just needs a bit of experience and game management. I'd say Shane has put down a bit of a marker after that performance.
"All's well that ends well, we got the bonus point, which at half-time we would have suggested we would have got, but Newcastle made it very hard for us.
"We lost complete control of the game in the second half, in a 20 or 30 minute period, we couldn't get a scrum feed or a penalty and we couldn't get the ball off the pitch.
"They've got a world-class back line and they are a very skilful side and they had us on the ropes, and we are disappointed with that aspect."
Tries fromÂ Tonga Lea'aetoa, Robbie Russell and Seilala Mapusua before the break, plus two late Geraghty touchdowns wrapped up the win for Irish, with Flood and Ollie Phillips scoring for Newcastle.