The 20-year-old suffered the injury during the Swans' 4-0 win over Sunderland at the Liberty Stadium last Saturday.
"I went up for a challenge in their box early in the first half and I landed awkwardly," Davies told the club's official website.
"I managed to play on for nearly 30 minutes but I knew I had to come off. I am disappointed but I just have to do my rehab now and come back as soon as possible."
The Welshman has netted twice in 12 appearances for the Swans this season.