The Lyon midfielder suffered the injury during his side's UEFA Champions League play-off defeat to Real Sociedad last week and sat out Saturday's defeat to Evian as a result.
In a statement on their website, the French Football Federation confirmed that Grenier had "not recovered enough" from the injury and will miss the FIFA World Cup qualifiers on Friday and Tuesday.
France are a place and a point behind leaders Spain in Group I and will face the group's bottom two sides in the next week.
Franck Ribery is likely to feature against Georgia for his first international appearance since winning the UEFA Best Player in Europe award.
Head coach Didier Deschamps believes Ribery is sometimes guilty of overplaying but feels his quality is undeniable.
"When he's playing for his country he really gives his best and tries to be decisive and make the rest of the team give their best," he said.
"Sometimes he can play too complicated trying to make the difference. He just needs to find the good balance.
"At Bayern Munich he's been doing well for a while now. We can be proud of what he is achieving. He's a French player and the best European one."
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