Eaton left Friday's game at the Seattle Mariners with a side injury after one at-bat. He missed Thursday's game entirely after crashing into the right-center field fence Wednesday in the White Sox's series finale at home against the Texas Rangers.
Eaton spent time on the disabled list in May with a strained right hamstring and has dealt with a number of other nagging injuries this season to his leg, wrist and finger.
Eaton is batting .304 this season and has solidified the leadoff spot with a .370 on-base percentage that was eighth best in the American League at the start of play Saturday.
Danks had one previous stint with the White Sox this season from April 10 to May 7. He is batting .276 at Charlotte with 18 doubles, 16 home runs and 56 RBIs in 88 games. On July 31, he became Charlotte's all-time hits leader when he surpassed Joe Borchard's total of 439.