According to the club, the condition is uncommon for a pitcher.
The treatment is rest and nerve medication, which Padilla will start Tuesday.
The club said it is difficult to estimate how long the pitcher will be out.
The Dodgers' Opening Day starter was placed on the disabled list after his start last week in Cincinnati with discomfort that originally bothered him in Spring Training.
"When I first felt pain, my body was warm, and I feel more pain now, but the weather is colder now, too," said Padilla.
"In the past, in Spring Training, I was able to throw through the pain, but this is a little more serious. Since they haven't told me anything, I don't know if it's a tendon or muscle, and I don't know what to expect."
Padilla, signed last summer after being released by Texas, was key to Los Angeles' stretch run, going 4-0. He was re-signed as a free agent to a one-year deal for $5.025 million.
Padilla will likely be replaced in the rotation by Jeff Weaver once he returns from injury.
Ken Gurnick is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.