Swansea are rooted to the bottom of the table with 12 points and have scored 10 goals in 18 games and Bony's absence is a blow ahead of Saturday's clash with 14th-placed Crystal Palace at the Liberty Stadium.
"Hamstring injuries are normally a minimum of seven to 10 days, but we will have to see if it is worse than that," Clement told the club's website (www.swanseacity.com).
"He has injured his hamstring but how significant it is we are not sure."
The Ivory Coast international has been plagued by a series of injuries since his return to the Welsh side from Manchester City in the close season.
"It was a strange one," Clement added. "It was only two minutes into the game and I don't know what action he had done at that point to create the injury."
Midfielder Ki Sung-yueng missed the game at Everton after picking up a minor calf injury in training but Swansea are hopeful he will be available for the Palace game.
Swansea have one victory in 10 league games going into Saturday's game while Palace are unbeaten in seven games and on a two-match winning run after beating Leicester City in their last outing.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Raissa Kasolowsky)
