Jones had Tommy John surgery Tuesday. The operation was performed by Dr. James Andrews.
Jones also had back surgery May 5. The right-hander has been limited to just two appearances this season, failing to retire a batter in a pair of April games.
''Probably the last thing you're thinking of is he'd have an elbow issue, especially with the back and everything,'' manager Robin Ventura said. ''You feel bad for him.''
Jones went 4-5 with a 4.15 ERA in 70 games last season.
Chicago reliever Matt Lindstrom and outfielder Avisail Garcia are expected to begin rehab assignments Friday with Triple-A Charlotte. Lindstrom had surgery on his left ankle May 23, and Garcia also needed surgery after hurting his left shoulder attempting a diving catch April 9.
''You look forward to when they're healthy, more than sit there and wish they could have been there all year,'' Ventura said. ''It's not productive to do that.''
Garcia was initially thought to be out for the season, and Lindstrom's return was no sure thing either. If they can both make it back, it would be a boost to a Chicago team that was 5 1/2 games out of a wild card at the start of play Wednesday.
''They're good players. You get Lindy out there and add to the bullpen, and Avi playing - they're good players, and we're counting on them even leading to spring training,'' Ventura said. ''You get them back, you kind of feel a little more complete.''
Ventura was noncommittal when asked if Lindstrom could take over as the team's closer when he returns.
''We'll see when he gets back here, how it all lines up,'' Ventura said.