According to manager Tony La Russa, Greene is available to field or pinch-run.
"It is frustrating to not be able to bat," Greene said.
"I am just trying to get the swelling an inflammation down. It has gotten better every day."
Ryan has scuffled at the plate for much of the season, and entered Tuesday hitting .103 with a .121 slugging percentage over his past 20 games. He has played better since the All-Star break, however, reaching base three times in three starts while driving in two runs. He popped out in his first at-bat, ending the second inning.
Greene began play on Tuesday with a .313 average, a .389 on-base percentage and a .500 slugging percentage since being called up July 2.