"We'll re-evaluate at that time, and hopefully the pain and soreness have subsided from that time," said Padres manager Bud Black, who told reporters of Johnson's injury after Saturday's 3-3 tie with the White Sox.
"[Hopefully] we'll get him on a throwing program and get him back up to speed."
Johnson -- who wasn't available for comment -- was scheduled to start Sunday's Cactus League game in Surprise against the Rangers. It would have been his fifth start of the spring. In four starts, the 30-year-old has posted a 3.38 ERA.
Johnson last pitched Tuesday against the Mariners and complained of pain the following day, Black said.
"He's really frustrated -- in his words, he's devastated," Black said. "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."
Johnson had health issues last year with the Blue Jays, including inflammation in his triceps as well as a strained forearm. After the season, he had bone spurs and loose particles removed from his elbow.
"He's looked great; he's looked sharp," said Padres catcher Nick Hundley. "From day one, Josh came in here in great shape. He put in unbelievable work in the offseason. This is a definite blow to our guys, our staff and Josh."
The Padres will likely fill Johnson's spot in the rotation -- he was expected to be the fourth starter -- from within, with consideration given to right-handers Matt Wisler and Donn Roach and left-hander Robbie Erlin.
The Padres jumped on Johnson early in the offseason, giving him a one-year deal at a reasonable price with the hopes that a change of scenery, a pitcher-friendly ballpark and a clean bill of health would be well worth the risk for a pitcher who led the NL in ERA (2.30) in 2010.
As part of Johnson's contract, the Padres hold a club option of $4 million for 2015 if Johnson is unable to make at least seven starts, risk protection given his health issues last season.
Even before he got to Peoria last month, Johnson and pitching coach Darren Balsley were on the phone after Balsley immediately went to work on watching video of Johnson.
Last month, Balsley told MLB.com that he didn't think it would take much to get Johnson back to his old form.
"He had the one mediocre year, but everyone has a bad year," Balsley said. "And contributing to his trouble was arm problems. I expect him to be as good as he was before."
Since the start of the year, the Padres have lost left-handed pitcher Cory Luebke to a second Tommy John surgery and Joe Wieland will likely miss most of the first half of the season after it was announced Saturday that he'll have arthroscopic surgery on his right elbow. Wieland had Tommy John surgery in 2012. His elbow ligament is fine, though.
The depth the Padres have built in the starting-pitching ranks has already been tested.
"You want as much depth as possible," Black said.