The striker, a summer signing from Dutch side Vitesse, is sidelined along with Michu, leaving Michael Laudrup short of options in attack ahead of a trip to Manchester City.
Bony, who has scored seven goals for the club since joining from Vitesse this summer, was injured in the 1-0 Europa League defeat to Valencia on Thursday.
The striker is due to have a scan to determine the extent of the problem, but Laudrup is anticipating a long absence.
"It could be quite a serious one," Laudrup told reporters. "We will have to wait two or three days until we get a scan, but unfortunately I think we are talking several weeks at least."
Laudrup is also without Michu but is hopeful the Spain international will return to action in the coming weeks.
"It is a blow, but that's how it is. You have to accept it and when you have a lot of games this is a possibility," Laudrup said. "Let's see what happens with Michu in the next week or so."