The Tottenham defender suffered a blow to the face in his club's 1-0 Premier League defeat to Newcastle United on Sunday.
When asked about Chiriches' injury after the game, Tottenham manager Andre Villas-Boas said: "I think he broke his nose."
The Romanian Football Federation confirmed in a statement that Chiriches, who boasts 22 caps, would not be fit to feature in either match against Greece.
Dinamo Bucharest's Dragos Grigore has been named as a replacement for the 23-year-old.
Romania play the away leg of their tie with Greece in Piraeus on Friday, with the return match taking place in Bucharest four days later.
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