The Wales international suffered the injury during the first half of Saturday's 4-0 win over Sunderland, and was withdrawn shortly after the interval before undergoing scans to assess the full extent of the problem.
It means Davies is set to miss the two Europa League clashes with Kuban Krasnodar, as well as Premier League fixtures against West Ham, Cardiff and Stoke, while he is also a doubt for Wales' friendly against Finland.
"I went up for a challenge in their box early in the first half and I landed awkwardly," Davies told the club 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."