They can be activated at any time during camp, but must be back in action for the opener or they will be forced to miss the first six regular season games.
The team has not revealed the nuture of the problem, but it arose during activities off the field.
Johnson did miss some time this offseason with a back injury.
Linebacker Aaron Lynch (leg), wide receiver DeAndre Smelter (knee) and offensive tackle Trent Brown (unknown injury) were also placed on non-football injury, while center Daniel Kilgore was placed on the physically unable to perform list, recovering from a fractured leg.
Hills played for Dallas last season, while Wallace spent time with Miami. Reaves rushed for 78 yards last year.
Bell was facing a three-game suspension due to a substance abuse violation. Bell rushed for 1,361 yards and eight touchdowns last season.
Gurley was recovering from an ACL injury in his left knee, but he will be able to participate in training camp practices. He was the 10th overall pick in the 2015 draft.
In addition to Johnson, the team also signed offensive lineman Ronald Patrick and linebacker Justin Anderson.
Tucker spent the last two seasons with the team, but mostly on the practice squad. He did earn playing time in two games as a rookie in 2013, though with no rushing attempts.