Manager Pedro Grifol said Thursday that Jimenez (hamstring) will begin a rehab assignment next week, Vinnie Duber of AllCHGO.com reports.
Jimenez was initially given a one-month return timeline when he landed on the IL on May 22 with a left hamstring strain, but he appears to already be approaching a return to game action. It's unclear exactly when the 27-year-old will head to the minors or how long he'll remain there, but the White Sox will likely be eager to return him to their lineup as soon as possible given their offense's poor performance to start the year.