It was not immediately known if any of the coaches would decline that offer.
Both general manager John Mozeliak and Matheny said on Thursday that they hope Miller's replacement will come from within the organization. No specific reason was given for Miller's dismissal.
"Just from a standpoint of when we were putting the coaching staff together last year, we were doing it quickly and Dyar deserved that opportunity," Mozeliak said. "But when we look at this long-term, we want Mike to be able to put his fingerprints on it as well."
Miller joined Matheny's staff in January, after Derek Lilliquist moved into the role of pitching coach. Though it was Miller's first season on the Major League coaching staff, the 66-year-old worked in the Cardinals organization for 20 years. Before serving as the Cardinals' bullpen coach, Miller spent four years as the organization's Minor League pitching coordinator.
The Cardinals' internal search for a replacement will likely include Brent Strom (Minor League pitching coordinator), Blaise Ilsley (Triple-A Memphis pitching coach) and Bryan Eversgerd (Double-A Springfield pitching coach).
"We were excited to have Dyar with us and are happy to give another person an opportunity, hopefully a guy within the organization," Matheny said. "I know there are some talented people and there's an opportunity to give them exposure to what's going on here. It's just going to be whoever fits the best."