The 20-year-old Wales international sustained the injury during Saturday's 4-0 win over Sunderland and, after eventually limping off, will undergo intensive treatment in the hope of returning in late November.
Davies told the club website: "I went up for a challenge in their box early in the first half and I landed awkwardly. 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 news is expected to result in Neil Taylor, who replaced Davies against the Black Cats, stepping up for a run of starts at left-back.