Bodmer suffered a left calf injury and Pouplin a left thigh problem in their 2-1 season-opening loss at home against Monaco at the weekend.
The Riviera side are already without defender Romain Genevois with a fractured shinbone and Swedish midfielder Niklas Hult, recovering from a hamstring problem.
The Nice defence will also have to do without the services of suspended Olivier Boscagli after his sending off last weekend for their trip to Troyes next weekend.