The defender suffered a calf problem in United's 3-0 Premier League defeat to Chelsea at Stamford Bridge.
Debuchy joins a lengthy injury list, with the likes of Fabricio Coloccini, Cheik Tiote, Papiss Demb Cisse and Yoan Gouffran also sidelined.
The 28-year-old attempted to play on after being kick on the calf, but he was withdrawn before the break.