The 29-year-old limped off in the Swans' 3-1 defeat to Everton on Monday, after only four minutes due to a hamstring strain, and he is expected to be out for seven to 10 days.
The former Manchester City striker will miss Saturday's home clash against Roy Hodgson's side and the trip to Anfield on Boxing Day.
"He's got a hamstring strain that he picked up at Everton and it will be about seven to 10 days, so we will see how we go with that," Britton said in his first press conference after he was confirmed as Paul Clement's replacement on Thursday.
On whether the marksman stands a chance of featuring against Watford on December 30, the player-coach said: "Maybe he has an outside chance, but that might be pushing it."
Bony has seen his second spell at the Liberty Stadium outfit marred with injuries. He has been restricted to 653 minutes from 11 league appearances, scoring only two goals for his bottom-placed side.