Gregorius, who will turn 25 in February, hit .226 with six homers and 27 RBIs over 80 games with the D-backs last season. He was hitting .310/.389/.447 in 57 games with Triple-A Reno to begin the season before injuries forced him into the D-backs' lineup.
Though he's yet to his stride at the plate in the big leagues, Gregorius is regarded as a solid defender with potential to improve at the plate. He showed flashes of that offensive potential during stretches with the D-backs, but has posted a .243/.313/.366 batting line in 191 games over parts of three Major League seasons.
As for Greene, he leaves the Yankees after an impressive rookie season in which he went 5-4 with a 3.78 ERA over 15 outings (14 starts).
With Greene's departure, the Yankees have Masahiro Tanaka, CC Sabathia, Ivan Nova and Michael Pineda as their top four remaining starters. All four enter the season with injury concerns, particularly Nova, who will be out until at least May as he continues to recover from Tommy John surgery.
The Yankees could still re-sign Brandon McCarthy, Hiroki Kuroda or Chris Capuano, or look elsewhere to round out the rest of the rotation in Greene's absence.
Paul Casella is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.