The team placed Bradley, who has been pitching for Triple-A Reno, on the seven-day disabled list Tuesday with a mild flexor strain in his right elbow.
The hard-throwing 21-year-old right-hander is considered one of the top young pitchers in the minor leagues and it was widely expected he would be called up by the major league club this season.
Bradley, the seventh overall pick in the 2011 draft, struggled in Reno, going 1-4 with a 5.18 ERA.
Diamondbacks general manager Kevin Towers said results of the MRI and CAT scan on Monday were ''good news'' because the problem is not that serious.
''The key thing is, structurally he is fine,'' Towers said. ''Most of our pitchers have some sort of discomfort when they throw. Small tears. That's just the wear and tear of throwing the ball 90-plus every fifth day for a whole season. ''
Towers said that hopefully the swelling subsides in a couple of weeks and ''he doesn't have the issue any more. I wouldn't be surprised at some point in his career he has tenderness in his elbow. That happens with pitchers.''