The France forward was dismissed in the first half of the champions' 4-3 Barclays Premier League win at Carrow Road for an alleged headbutt at Sebastien Bassong.
Manager Roberto Mancini indicated after the game on Saturday that an appeal would be lodged but, after further thought, the club have now announced there will not be a challenge.
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."
Nasri reacted angrily to a challenge from Bassong in a manner that was seen as violent conduct.
Mancini maintains the dismissal was harsh but conceded, as he met the media early on Monday to preview the Stoke game, an appeal may not succeed.
Appealing would have allowed Nasri to face Stoke as, due to the holiday season, a hearing would not have been heard until Wednesday. But by doing so the club would run the risk of having the ban extended if the appeal was deemed frivolous.
With Yaya Toure, Kolo Toure and Abdul Razak now due to depart for African Nations Cup duty with Ivory Coast after the Stoke game - rather than after Saturday's FA Cup tie against Watford as first thought - it was considered too much of a gamble.
Mancini said: "We don't know what can happen if we appeal. Usually they reject an appeal. This could be a problem because after this game we lose three players - Yaya, Kolo, Razak and we don't want to lose another."