Clark was forced off during the first half of Villa's 2-1 Premier League victory over Tottenham earlier this month and missed last weekend's FA Cup semi-final triumph over Liverpool.
And the 25-year-old now appears to set to miss the final with the issue.
Manager Tim Sherwood had been pessimistic about Clark's chances of a return, saying before the semi-final: "It's a real blow for Ciaran but that's the nature of the game – when you're in there, there's a chance of getting injured.
"He's certainly one that doesn't hold back and puts his body on the line. Unfortunately, he's caught one there and we'll see him next season."
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