The Dutchman has not featured since coming on as a substitute in the 1-1 draw against Chelsea at the end of January, and suffered the injury in training prior to the home defeat to Norwich.
And, after a scan on the problem, the 25-year-old underwent a small operation on Tuesday which will keep him out of action for three to five weeks, meaning that at the very least he will miss the Premier League games against Wigan, Manchester City, Fulham and Everton.
A statement on the Swansea website read: "Swans midfielder Kemy Agustien faces at least three weeks on the sidelines after undergoing an ankle operation."
Agustien has made 15 appearances so far this term, having also been troubled by hamstring problems earlier in the campaign.
But he has found himself largely limited to bench duty, initially by the form of Mark Gower and more recently by the loan signing of Gylfi Sigurdsson.