Manchester City midfielder Nasri was given a three-match ban.
He had a prolonged row with the media throughout Les Bleus' campaign in Poland and Ukraine which culminated in a loud, expletive-laden exchange with a journalist following the quarter-final defeat to Spain in Donetsk.
Nasri will miss the August 14 friendly against South American champions Uruguay in Le Havre as well as two World Cup qualifiers – away to Finland on September 9 and at home to Belarus four days later.
Paris Saint-Germain winger Menez has been handed a one-match ban for his behaviour during the last-eight loss to the eventual winners in the Donbass Arena, in which he clashed with referee Nicola Rizzoli and his own team captain Hugo Lloris.
However, both Newcastle United winger Hatem Ben Arfa and Rennes midfielder Yann M'Vila escaped punishment for - respectively - an alleged argument with Laurent Blanc and not shaking hands with the then manager or replacement Olivier Giroud during the same match.