Jones, who was a candidate to take over the closer's job during spring training, was first diagnosed with a gluteus-muscle strain in spring training, then a sore hip before the back problem was identified.
He made two appearances during the first week of the season but struggled, giving up a combined four earned runs without recording an out, which left him with an ERA of infinity.
Jones, 28, underwent a microdiscectomy, where a small portion of bone is removed to alleviate nerve irritation.
“Everything went very smoothly and very well,” White Soxgeneral manager Rick Hahn said. “What they saw was as expected, and we’ll reevaluate in a month. After a month, it’s conceivable we’ll ramp up baseball activities and he can return soon thereafter as tolerated by the baseball activities. At that point it’s more about getting him into pitching shape than dealing with the back."