Kinney, who underwent Tommy John reconstructive elbow surgery last March, has been sidelined from his recovery program, and a timetable for his return to active duty is unavailable at this point.
"I don't know if I would say it's a big deal yet," general manager John Mozeliak said on Friday. "I would definitely say it's slowed down his progress.
"He had some discomfort, and instead of having him pitch through the pain, we wanted him to be looked at. We're going to take a conservative approach and allow him to take a step back from his throwing program and work on his rehab."
On the brighter side, reliever Tyler Johnson has begun throwing off a mound. Johnson sustained a rotator cuff strain and tendinitis in his left shoulder during Spring Training, and has not pitched in a game this year.
Johnson has been on a throwing program, and because it has gone well, he has moved forward to throwing bullpen sessions. A date has not been set for him to move up to facing hitters, but the hope is that his advancement could be relatively quick from here on out.
Mozeliak said it would be inaccurate to compare Johnson's situation to the start of Spring Training, indicating the lefty could conceivably be back in less than six weeks.
"I think he's a little further along than that, because all the month of March, and three weeks into April, he's been able to condition pretty much every other part of his body," Mozeliak said. "I would speculate that, if all goes well, this would be an accelerated version of Spring Training. But we don't want to rush him either."
Matthew Leach is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.