The Ivory Coast international suffered the blow just five minutes into the 3-1 defeat to Everton on Monday (18 December) at Goodison Park and was replaced by Tammy Abraham.
And though the strain will keep him out for only two weeks it is enough to ensure he will miss the visit of Palace on Saturday and the trip to Anfield to face Liverpool 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," said Britton (via Sky Sports), installed as the temporary replacement for Paul Clement, who was sacked on Wednesday.
Asked whether the injury could see Bony miss a third game, with the trip to Watford following on 30 December, the Swans midfielder added: "Maybe he has an outside chance, but that might be pushing it."
Swansea are rooted to the bottom of the Premier league after successive defeats and the string of fixtures against fellow strugglers Palace, Liverpool and Watford will go a long way to determining whether they can retain their top-flight status.
But while Bony – who has netted just three goals since returning to the Liberty Stadium in the summer – is sidelined for the games against Palace and Liverpool, Swansea may be able to call upon defender Kyle Bartley, who has missed the last four months with a knee injury.
The 26-year-old has only played three times this season and not since the Carabao Cup win over Milton Keynes Dons in August but after returning to training played in the International Cup defeat to Athletic Bilbao B earlier this week.
"It was good to get back out there," he said, according to the official Swansea website. "To get an hour under my belt this week should put me in good stead to hopefully get back in the first-team squad.
"It was important for me to get the minutes in my legs and make sure all is okay. I think I ticked a lot of boxes with the game this week.
"I have been training for a couple of weeks with the physios and fitness coaches. They have done a great job with me behind the scenes.
"We now go into a busy Christmas period and hopefully I can help the team out. I feel ready to come back before the new year.
"I was devastated to get injured. I felt I had a good pre-season and had made an impact in the games I played, so it has been frustrating to be on the sideline.
"Unfortunately, these things happen in your career and they help you become stronger mentally and physically. I think I can come back as a better player and person."
A more pressing concern for Swansea, however, is filling their managerial position after Clement was dismissed after less than a year in charge, after taking the south Wales club to the foot of the table.