The sagittal band is part of the extending mechanism of a finger. The injury is causing a "dislocation of the exterior tendon," according to a release by the team.
Anderson, 29, is hitting .301 this season, with six home runs and 25 RBIs.
The White Sox are 2½ games behind Cleveland in the American League Central entering Thursday's games. They are also 2½ games back in the AL wild-card race.