The Spaniard sustained the injury during Saturday's 4-2 Premier League defeat at Chelsea, failing to emerge for the second half.
After undergoing a scan on Sunday, Amat saw a consultant who confirmed he would be missing until the end of October.
The issue will come as a big blow to the 22-year-old, who has kept his place in the Swansea starting XI despite the arrival from Napoli of Argentina international Federico Fernandez - Amat's replacement at Stamford Bridge.
Fernandez is now likely to get his chance to shine in the first team, while Kyle Bartley should regain his place on the substitute bench.
Swansea welcome Southampton to the Liberty Stadium this weekend.
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