The left-back sustained an ankle injury in the club's 4-0 win over struggling Sunderland and now looks set to miss the games against Kuban and West Ham
The left-back was withdrawn shortly after half-time, having fallen awkwardly in the first half, and manager Michael Laudrup admits the Welshman is a doubt for Thursday's European clash with Kuban Krasnodar and next Sunday's Premier League fixture against West Ham.
"It's a twisted ankle," the Dane said. "I think he just fell in the first half, so at half-time we agreed that he should start the first couple of minutes, but already after two or three minutes you could see that it wasn't good.
"So we changed him and we have to see tomorrow (Sunday) or the day after how bad it is.
"I'm not the doctor and we have to wait for the X-ray, but probably he will be out for the next two games at least."
Davies has scored twice for Swansea in the Premier League this season, but missed Wales' two FIFA World Cup qualifying fixtures against Macedonia and Belgium with a hamstring problem.
Home fixtures against Sunderland have become something of an unhappy occasion for Swansea's full-backs, with Neil Taylor missing all of last season after fracturing his ankle in last September's meeting.