Gaines, 26, was carted off the field Sunday during while defending a pass in practice.
“It’s football. I hate to lose him, but at the same time, we know that those things do happen,” head coach Hue Jackson said.
Gaines signed with the Browns in March as a free agent after playing last season with the Buffalo Bills.
Gaines was rated the NFL’s 13th-best cornerback by Pro Football Focus last season and had 59 tackles with one interception and three forced fumbles in 11 starts.
The Rams selected Gaines in the sixth round of the 2014 draft. He has played in 37 career games (36 starts) and has 3 interceptions, 31 passes defended, 4 forced fumbles and 185 tackles. He missed the entire 2015 season because of a foot injury.