Vidal Nuno, the departing piece of Sunday's deal for McCarthy, was penciled in to start on Monday night in Cleveland, and it initially appeared as if callup Bruce Billings would take his place.
But before Sunday's game, Girardi said that Billings will not start on Monday, and that the eventual starter might not be on the current 25-man roster.
"It's possible," Girardi said.
After the game, the manager announced that Greene will start. The right-hander has gone 5-2 with a 4.61 ERA in 13 Triple-A starts this year.
Does Girardi envision Greene sticking around for more than one Major League appearance?
"Every opportunity is an opportunity to shine and get more opportunities," Girardi said.
Barring injury, McCarthy will get plenty of chances to stick. He brings a 3-10 record and 5.01 ERA to the Yankees, but has been much more solid in his two most recent starts (2-0, 2.13 ERA). Can he keep it up?
"He's got too good of stuff not to," Girardi said.
Alex M. Smith is an associate reporter for MLB.com. Follow him on Twitter @asmiff. This story was not subject to the approval of Major League Baseball or its clubs.