Padres general manager Josh Byrnes said Friday that Kelly would have surgery on his pitching arm during the first week of April.
Kelly is expected to miss 12 to 18 months while undergoing rehabilitation after the procedure.
The 23-year-old Kelly was 2-3 with a 6.21 ERA in six starts last season and twice was shut down because of elbow pain. He was a candidate for the rotation this year.
Kelly says doctors found small tears in his elbow. He'd felt pain during an outing on March 7.
"It's unfortunate, but it's something that has to happen," he said. "I'm just looking forward to putting this injury behind me and coming back strong in 2014."