D-backs manager Chip Hale said Wednesday that Butcher would have input on who the new bullpen coach would be.
While it's possible that the team could fill the position internally, it appears that it's more likely that Stottlemyre's replacement will come from outside the organization.
As far as filling Green's spot goes, there are three names to keep an eye on: former third base coach Matt Williams, Tony Perezchica and Phil Nevin. All three have ties to the organization and could fill Green's role not just at third, but also as the team's infield coach.
Williams was the D-backs' third base and infield coach before becoming the Nationals' manager in 2014. Extremely popular in the clubhouse during his time with the D-backs, Williams was dismissed as Nats manager earlier this month.
Perezchica has been a coach or manager in the Minors since 1997, first in the Yankees' system and, since 2003, with the D-backs. Over the past two seasons, he has been Arizona's field and infield coordinator and is well respected for his work with the organization's infielders.
Nevin has managed in the Minors for the past seven seasons, including 2014-15 for the D-backs' Triple-A Reno team. A 12-year big league veteran, Nevin played third base, first base, catcher and outfield. Over the past two years, he has interviewed for several Major League jobs, and the D-backs think he has nothing left to prove in the Minors.
Steve Gilbert is a reporter for MLB.com. Read his blog, Inside the D-backs, and follow him on Twitter @SteveGilbertMLB. This story was not subject to the approval of Major League Baseball or its clubs.