The 25-year-old, who has only recently returned from two months on the sidelines with a troubled hamstring, suffered the problem in Saturday's 3-1 defeat to Stoke City.
He joins a growing Black Cats absentee list, which also features Lamine Kone, Wahbi Khazri, Didier N'Dong, Jordan Pickford, Jan Kirchhoff, Victor Anichebe, Steven Pienaar, Duncan Watmore, Paddy McNair, Lynden Gooch and Lee Cattermole.
"Jack had a medial ligament injury," Moyes told Sky Sports News. "He was scanned on Monday after getting it in the game on Saturday, and the scan has not come up with anything. I'm hoping that it shouldn't keep Jack out too long.
"He's just beginning to get a couple of games under his belt. To have Jack out [against Burnley] was a blow.
"They thought that he'd opened his medial ligament so the scan has not shown a great deal. We're hopeful that he might just have to stay off it for a week or so and let's see how he is."
Sunderland currently sit second from bottom in the Premier League table, on 15 points.