"We'll get more details in the next couple of days, [but] it doesn't look real good for him," Bears coach Matt Nagy said after the game.
Smith made contact with Cowboys running back Ezekiel Elliott as he crossed the goal line to cap Dallas' opening 17-play, 75-yard touchdown drive. Kevin Pierre-Louis replaced Smith on defense for the remainder of the game.
Chicago's first-round pick in 2018 (eighth overall), Smith had a career-best 15 total tackles last week. The Bears are already without another starting inside linebacker, Danny Trevathan (elbow), who was replaced in the starting lineup by Nick Kwiatkoski.
Smith entered Thursday's game with 99 tackles, two sacks and one interception this season. As a rookie last season, he played in all 16 games and posted 122 tackles, five sacks and an interception.
"It could be a big loss," Nagy said. "The way that Roquan has been playing the last couple of weeks, he's been flying around and making plays. You just love that confidence that he's bringing to the defense.
"The depth that [the front office] has created on their roster allows guys like Kevin Pierre-Louis to step up and make plays. You feel bad for Roquan, but we know we have guys that can fill in. It's a good thing to have that depth."
Pierre-Louis finished the game Thursday with five tackles, two pass breakups and one tackle for loss.
The Bears (7-6) next travel to Green Bay for a showdown with the Packers in Week 15.