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Thornton was saddled with the loss Monday after allowing two runsone earnedin the top of the eighth.
Thornton threw a scoreless inning against the Indians on Sunday to earn his fourth hold of the season.
Thornton felt good after his 40-pitch live batting practice session Wednesday, CSN Chicago reports.
Thornton (elbow) threw a 35-pitch bullpen session Sunday without discomfort, Dan Hayes of CSNChicago.com reports.
Thornton had an MRI that showed inflammation in his left elbow but no other damage, ESPN Chicago reports.
Thornton ended a bases-loaded threat by the Twins on Saturday, recording the final out to seal the win for the White Sox and propel Thornton to his third save of the season.
Thornton pitched a scoreless 10th inning on Friday, allowing one hit and one walk with one strikeout, to earn the win against the Angels.
Thornton allowed one hit over 1.1 scoreless innings to record his second save of the season Saturday. He struck out two.
Thornton will revert to a setup role with Addison Reed earning the club's official closer title, the Chicago Tribune reports.
Manager Robin Ventura announced the White Sox will return to a closer-by-committee situation with Chris Sale returning to the rotation, ESPN Chicago reports.
Thornton allowed two runs on two hits, one of which was a one-out, two-run, walk-off home run in the bottom of the ninth inning of a 5-4 loss to Detroit Friday night.
Thornton picked up a four-out save on Sunday against the Red Sox, but he doesn't necessarily have the closer's job from Hector Santiago yet, CSN Chicago reports.
Manager Robin Ventura said Thornton will maintain his eighth inning role, the Chicago Tribune reports.
Thornton's eighth-inning appearance does not eliminate him from consideration for the closer's role, CSN Chicago reports.
White Sox manager Robin Ventura said that he has decided who his closer will be but won't announce it yet, the White Sox official site reports.
Thornton struck out two Dodgers over a perfect ninth Thursday to pick up his first save of the spring.
Manager Robin Ventura has yet to tip his hand in the White Sox's closer battle, which means Thornton's role for 2012 is still not official, the Chicago Sun-Times reports.
The White Sox's closer battle appears to have narrowed to Thornton and Jesse Crain, the Chicago Sun-Times reports.
Manager Ozzie Guillen said the race for the White Sox's closer role might be more open than previously stated, the Chicago Sun-Times reports.
Manager Robin Ventura reiterated Thursday he will go into spring training with Thornton as his closer, but that could change before the beginning of the season, the White Sox's official site reports.
Manager Robin Ventura said Thornton could get the first crack at the White Sox's closer role in the wake of Tuesday's Sergio Santos trade, the Chicago Tribune reports.
Thornton tossed two scoreless innings to earn his third save of the season Saturday against the Cubs.