Bartley was injured in the Swans' Carabao Cup game at MK Dons on August 22 and was initially expected to be sidelined for six to eight weeks.
But scans revealed the problem was more serious than first feared and the 26-year-old had an operation this week.
Paul Clement told his club's official website: "We were disappointed to lose Kyle. He has now had surgery and we expect him to be out for around 12 weeks.
"The period of 12 weeks is an estimate, and we will have to see how Kyle responds to his rehabilitation."