Johnson suffered a laceration to his liver during Wednesday's practice and will be evaluated on a week-to-week basis to determine when he can rejoin the Browns on the field.
He was injured when rookie tight end Harrison Bryant landed on him during a red zone drill, after Johnson dove to break up a pass.
"[We] would've preferred that he held back instead, out of an abundance of caution, and not dive for it. But they kind of both dove, and Harrison landed on top of him," coach Kevin Stefanski said this week. "Really unfortunate, unlucky."
Johnson, who signed with the Browns on a one-year deal this offseason, was projected to be the team's starting nickelback coming into training camp.
Chubb has been in concussion protocol since Monday, when he was brought down with a horse-collar tackle during practice.