The France forward was sent off in the first half of the champions' 4-3 Premier League win at Carrow Road on Saturday following the incident with Bassong.
Manager Roberto Mancini was furious about the dismissal after the game and indicated that an appeal would be lodged.
But after further thought, the club have now announced there will not be a challenge.
Nasri will therefore be banned for three matches.
A statement from the club read: "The Football Association has confirmed that Samir Nasri will serve a three-match suspension, following his red card during the 4-3 win at Norwich on Saturday.
"Nasri will now miss the New Year's Day visit from Stoke City, the FA Cup tie with Watford (January 5) and the trip to Arsenal on January 13."