It means Lee might be finished with the Cowboys, Moore said.
Lee's contract runs through 2019, with a $7 million base salary and a $10.1 million salary cap hit for next season. The Cowboys can save $7 million by cutting him before next season, and Lee would count only $3.075 million in dead money.
The Cowboys have rookie linebacker Leighton Vander Esch, who they drafted in the first round (19th overall) ready to step in.
"We anticipate (Lee) to be out for a little bit, no question about that," head coach Jason Garrett said.
The 32-year-old Lee missed five games last season and already has missed three this year because of a pulled left hamstring.
The 6-2, 245-pound Lee has missed 45 of a possible 136 games in his career, 13 because of hamstring issues, according to Moore.
Lee was drafted by the Cowboys in the second round (No. 55 overall) out of Penn State in 2010. He was selected to play in the Pro Bowl in 2015 and 2016, and was chosen first-team All-Pro in 2016.
In his career, Lee has 695 tackles, three sacks, two forced fumbles, four fumble recoveries and 13 interceptions, two of which he returned for touchdowns.