Greene (groin/back) will begin a rehab assignment at Triple-A Louisville on Tuesday, Charlie Goldsmith of Fox 19 Now Cincinnati reports.
Greene felt good coming out of a 20-pitch simulated game Wednesday, and after completing a bullpen session Saturday without issue, the right-hander will get the green light to pitch in a minor-league game. If all goes well Tuesday, the 25-year-old could be activated from the 15-day injured list to start the Reds' final game before the All-Star break July 13 versus the Rockies. Otherwise, he'll likely aim to return shortly after the second half of the season begins.