Greene, 29, was released by the Astros at the end of spring training and signed with the White Sox on April 1. In four seasons with the Cardinals and Astros, the right-hander has a career line of .224/.292/.356 in 689 plate appearances.
The White Sox took Sanchez in the Rule 5 Draft from the Angels and are required to keep him either on their active roster or on the disabled list. If they do not, they must offer him back to the Angels.
Sanchez played in one game, making two plate appearances, coming in after Gordon Beckham injured his hand against the Nationals and leaving three innings later in a double-switch. He grounded out in both plate appearances.
As the White Sox do not have have room on their 40-man roster, a corresponding move will need to be made to make room for Greene.
Update: Charlie Leesman is the one designated for assignment. Leesman has been in extended spring training after offseason knee surgery. I don't think he's going to get through waivers unclaimed, should that be the route the White Sox go with him.