"That's the primary position," Cashman said.
During a more than one-hour conference call with New York-based reporters, Cashman said it would have been easy to "plug and play" and stay with Girardi after a decade that included a World Series title and six playoff appearances.
But the overriding reason that Cashman soured on Girardi was the "connectivity and communication" issue with players, which was a concern to the GM as the Yankees move into their next phase of their youth movement led by Aaron Judge, Gary Sanchez and the rest of the Baby Bombers.
Appearing at the same time on WFAN, Girardi was not aware of Cashman's comments, but he was asked about his relationship with the players.
"I thought it was really good," he said, adding that he had received some "wonderful texts" from younger players after he was fired.
Girardi denied rumors that he did not get along great with closer Aroldis Chapman or Sanchez, and the former catcher said the "sky is the limit" for Sanchez.
"For me it was always encouraging him, trying to make him better and telling him how important he is to our team," he said, adding of Sanchez's defensive struggles that "we didn't have an easy staff to catch."
Cashman said he did not know if owner Hal Steinbrenner would sign off on his recommendation to let Girardi go until days before the decision was announced.
"We don't make changes at that level lightly," Cashman said.
Girardi said: "I thought I would be back."
He described getting fired by Cashman.
"It was quick and to the point," he said. "Brian said they had decided they were going in a different direction."
He added: "It happens. It has happened to people who have a lot more wins than me. I think Brian wants something different and he's going to go out there and try to find that. Joe Torre won an awful lot and they went out and found someone and that guy was me, and Joe has won a lot more than I have."
The GM also said he doesn't know what would have happened if Girardi had led the Yankees to the World Series this season. They lost in Game 7 of the ALCS. In the weeks leading up the Yankees' final game, Girardi talked about having to speak with his family to decide if he would want to return or not. The decision, though, was ultimately, taken out of his hands.
Cashman said Girardi's failure to call for a replay that largely cost the Yankees Game 2 of the ALDS was not a factor in the decision.
While Cashman didn't think that Girardi was communicating well enough with the clubhouse, Cashman disagreed with reports that he and Girardi went separate way at times over analytics or that the two had a deteriorating relationship. Cashman called Girardi an "exceptional manager."
Girardi echoed that he got along fine with Cashman.
"Brian took a chance on me," he said. "There were guys who wanted someone else. He took a chance on me, so I'll be forever grateful."
Cashman said he has no timetable for naming Girardi's successor and had no preconceived candidate or list of replacements when he informed Girardi of the firing.
"Whoever the lead horse will be, hopefully they will be pretty obvious and they will win by 16 lengths, like Secretariat did," Cashman said.
While Cashman would like to put the person in place as soon as possible, he is not in a rush with no other vacancies in Major League Baseball. Cashman is not looking for the anti-Girardi.
"It is an easy narrative to run to the opposite of what you already had," Cashman said. "I think if someone was structured and demanding, then you want to go from an old-school, heavy-handed personality to a new school, players manager. That's not a narrative I'm falling into. We are looking for the best person possible that can assist in executing our strategy."
The Yankees will conduct interviews and then make each candidate available to the media. Sources have said that the Yankees will seek an "A.J. Hinch-type," but Cashman, at least publicly, talked in broader terms of an ideal candidate checking a lot of boxes, including communication, being analytical and having a good grasp of handling the media. These are all qualities that have been associated with Hinch.
Girardi acknowledged that he probably didn't give the media everything they wanted.
"The thing about my personality is that I'm very protective of the players," he said. "And you have to be these days with social media."
Cashman said he does not have to have a pre-existing relationship with any future manager. Asked specifically if Alex Rodriguez would be a candidate, Cashman declined to answer the question, not wanting to give a thumbs up or down on any individual.
"I do believe we are going to find someone we are going to be comfortable with that can lead us into the next phase of where we are with this franchise," Cashman said.
Cashman's contract also is expiring and he said the team is open to a new deal for him. He has been the GM since February 1998.
"I have had a few conversations with Hal Steinbrenner," Cashman said. "They haven't been concluded yet."
As for Girardi, he said he will probably go back to broadcasting for a while, but he "absolutely" wants to manage again.
"I want to manage again because I love the competition," he said.
Information from The Associated Press was used in this report.
Olney not surprised about Girardi
Buster Olney joins SportsCenter to break down the Yankees' decision to part ways with Joe Girardi after 10 years as the franchise's skipper.