In two rehab starts, Greene has not allowed a run in 6 1/3 innings while recording 12 strikeouts. He most recently pitched Wednesday with Triple-A Toledo and gave up four hits while retiring 10 batters before joining the Tigers in Oakland.
Bench coach Gene Lamont said that Greene might make one more rehab start before a decision is made.
The issue, according to Lamont, is whether Greene's blister issues will continue the longer he pitches in a particular game. Green threw 97 and 85 pitches, respectively, in the two starts before a shortened outing against Cleveland before going on the DL.
"Is the finger problem going to be a problem if he throws 80 or 90 pitches?" Lamont said. "We don't really know that. That thing took an awful long time to heal. We just have to see how it plays out.
"I don't make those decisions. He's got a great arm. I think he can be a starter or reliever. The thing we just have to decide is where is best for him and where is best for the team."
Michael Wagaman is a contributor to MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.