"He's really frustrated -- in his words, he's devastated," Black told reporters, according to MLB.com. "He felt so good coming into December and into January. He felt really good coming to spring training. He felt great up until his last outing. Hopefully, it's nothing more than a strain."
The oft-injured Johnson was projected to be the team's fourth starter. He figured to pitch the first game of a three-game series against the Miami Marlins on April 4.
Black said Johnson will be re-evaluated in 10 to 14 days.
The Padres signed Johnson to a one-year, $8 million deal in November with performance bonuses based on games started. The 30-year-old was on the disabled list for part of 2013 and had surgery on Oct. 1 to remove bone spurs and loose cartilage in his right elbow.
Johnson, who spent the first eight years of his career with the Marlins, led the NL with a 2.30 ERA in 2010. He went 2-8 with a 6.20 ERA in 16 starts for Toronto last year.
In his spring debut March 3 against the San Francisco Giants, he struck out two in two hitless innings.
"Everything was comfortable," he said. "I wasn't fighting myself or my body."
Information from ESPN.com's Jerry Crasnick and the Associated Press was used in this report.