The Dutchman was hurt during Wednesday's FA Cup third round replay defeat against Arsenal at the Emirates.
He is due to miss Swansea's next five games, starting with Saturday's home game against Stoke and including the second leg of the Capital One Cup semi-final against Chelsea.
"We had bad news after losing the other day, we have lost Kemy Agustien for, I think, the next four weeks with his ankle," manager Michael Laudrup said.
"He twisted it in the first half, he tried to start the second half and there was some problem with the ligaments.
"It could have been a lot worse, but still four weeks means he will be out for at least our next five games, four in the league and the game on Wednesday (against Chelsea) as well."