Bradford was injured in the first quarter of Saturday night's 33-14 preseason victory at Cleveland. He landed hard on the knee when hit by Browns defensive end Amonty Bryant after throwing a pass.
ESPN and the St. Louis Post-Dispatch both reported the extent of the injury.
Coach Jeff Fisher is scheduled to meet with reporters on Sunday evening.
Bradford, the first overall pick of the 2010 draft, missed the last nine games last season after getting injured at Carolina. Veteran Shaun Hill has been the backup and the Rams also have rookie Garrett Gilbert and Austin Davis on the roster.
After the game, Fisher thought Bradford might have hyperextended the knee.