The quarterback was injured in the fourth quarter of Sunday's 30-15 loss to the Carolina Panthers when he landed on his knee after being shoved out of bounds by Panthers safety Mike Mitchell.
The team offered no update on the extent of the injury and Bradford was not made available for comment.
He was to accompany the team on the flight back to St. Louis late Sunday afternoon.
Fisher said Bradford "was in significant pain on the sideline."
Bradford did have X-rays on the knee before leaving Charlotte.
"It's a left knee and it's the knee with the brace on it," Fisher said. "We're going to get off the airplane and he'll go have an MRI so we'll have some information on that later."
As for the injury, Fisher said "I'm not going to speculate. I'm not going to deal with hypotheticals until I find out what's going on with him. It makes no sense to rush to any conclusion with him. Not until we know."
Bradford was in pain on the Rams sideline for several minutes holding his left knee before two Rams assistants helped him to his feet and carried him to the bench. He was then helped to a cart and taken to the locker room.
He was replaced by Kellen Clemens.
Clemens said the Rams are "all thinking about 'ifs' right now", but said he will be ready next week if needed.
The Rams host the Seattle Seahawks next Monday night.
"It's never easy when you see a teammate go down," Clemens said. "When you see a person that you spend as much time with as I do with Sam and genuinely care about as a person. ... I have great respect for what he does on the field, but just as genuine concern for him as a person. It was not easy to see. Hopefully it's quick."
The Rams players were hoping for the best, but fearing the worst after the game.
"Sam is a really good guy and a smart player," linebacker Jo-Lonn Dunbar said. "That's really tough. I hope it's not as bad as it looks. It's tough to see him carted off like that."
Said center Scott Wells: "Even if Sam wasn't hurt, we still have some heavy lifting to do. But this adds extra to it."
Panthers quarterback Cam Newton said, "My heart goes out to Sam Bradford."
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