Bradford suffered the injury as he scrambled toward the sideline on a second-down play with 5:27 left. Carolina safety Mike Mitchell pushed Bradford before he stepped out of bounds, and the quarterback landed awkwardly on the knee as he fell to the ground.
Rams coach Jeff Fisher spoke to Bradford after the game but didn't want to speculate on the nature of the injury before further testing.
"I'm not going to deal with hypotheticals right now until I find out what really is going on with him," Fisher said. "It makes no sense to rush to a conclusion until we know. We'll get a medical, and then we'll go from there. He was in significant pain on the sideline."
Bradford was on crutches with the knee in a large, black, ankle-to-thigh brace as he left the locker room. According to Fisher, all his quarterbacks are required to wear a brace on the left knee anytime the team is wearing pads.
At the time of the injury, Bradford was 21-of-30 for 255 yards with a touchdown and an interception. Kellen Clemens replaced Bradford and was 2-of-4 for 19 yards.
Clemens is the only other quarterback on the Rams' roster or practice squad.
"I think we're all talking about 'ifs' right now," Clemens said. "Obviously, if called upon this week after a week of practice and an opportunity to work with the guys, we'll be ready to go."
Information from ESPN.com's Nick Wagoner was used in this report.