It would be a homecoming for Williams, who was on the D-backs' coaching staff from 2010-13 before serving as the Nationals' manager in 2014-15. He was dismissed following the 2015 season.
Alston, a 10th-round pick by the Rockies in the 1992 Draft, appeared in six games for Colorado in 1996.
Since retiring from playing in 2003, Alston has been with the A's organization, where he served as a Minor League coach and also as the team's rehab coordinator. Last season, he was promoted to Minor League roving pitching instructor for Oakland.
Williams would replace Andy Green, who was recently hired to manage the Padres, while Alston would take the place of Mel Stottlemyre Jr., who left last month to become the Mariners' pitching coach.
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.