The Diamondbacks fired Kirk Gibson prior to the end of the regular season.
It's unknown whether the Rangers have responded to the request. However, before the end of the season, Bogar said he was interested in only the Rangers job.
The Rangers interviewed pitching coach Mike Maddux and Triple-A manager Steve Buechele for their gig earlier this week.
General manager Jon Daniels said he would like to make a hire by the end of October.
Arizona, meanwhile, has received permission to interview the Oakland Athletics' Chip Hale, the Los Angeles Dodgers' Lorenzo Bundy, Kansas City Royals bench coach Don Wakamatsu and Chicago White Sox third-base coach Joe McEwing.