Letters were sent to the lightweight duo on Wednesday informing them of the decision to temporarily suspend them.
NAC president Bob Bennett confirmed the news on Thursday, and said a hearing on 24 October would determine if McGregor and Nurmagomedov would be hit with further bans.
The suspension that comes into effect on 15 October will last for at least ten days.
The NAC have withheld Nurmagomedovs fight purse for his decision to jump over the octagon and attack a member of the crowd.
Nurmagomedov confronted McGregors jiu-jitsu coach Dillon Danis just moments after submitting the Irishman in their lightweight title fight at T-Mobile Arena in Las Vegas.
McGregors purse was paid out in full, but video has caught the two-weight champion throwing one of the first punches in a brawl that erupted inside the octagon.
Nurmagomedov threatened to quit the UFC earlier on Thursday after it emerged the MMA promotion were going to fire featherweight Zubaira Tukhugov.
The Russian punched McGregor during the brawl, and climbed over the octagon to attack the 30-year-old.