On Monday, Perdue was relatively vague on where Glover's replacement should come from --in or out of state-- and when that person would be chosen.
In recent weeks, many analysts and observers have said it will take someone from the outside to clean up the North Carolina Highway Patrol, after months of resignations and firings.
However, North Carolina law doesn't allow for a national search. The patrol's commander must come from within the patrol.
But now, lawmakers on both sides of the aisle say they're interested in changing that law.
"I think if we got someone from outside with a lot of experience in law enforcement, but from another agency, that would be the way to go for the next person we hire," Republican Minority Leader Rep. Skip Stam said.
But the problem is unless there's a way to skirt the law that can't happen. Glover steps down in September and lawmakers don't meet again until January, which makes the governor's position on this somewhat confusing.
"I really don't want to call the General Assembly back for a special session, and I really don't want to wait until January," Perdue said.
Perdue says she'll appoint a task force this week to help find the state's next patrol commander.
That distance may work to her advantage. Her long-time friendship with Glover led to speculation that he got the job, because of that friendship.
When asked about it two weeks ago, the governor was resolute.
"I don't intervene in promotions," Perdue said on June 7. "I've never intervened in promotions."
But on Monday, Perdue backed off that statement by saying she has helped people advance their careers, but only those who deserve it.
"We've only recommended people for jobs who had the qualifications and the capacity to be employed gainfully," Perdue said. "So when you said you've never intervened, I have never intervened in terms of trying to find somebody who was unqualified or to put in somebody who couldn't meet the test of the job."
Just last week, it appeared Perdue was standing by Glover. He appeared with her and Crime Control Secretary Reuben Young at a news conference following her meeting with some 160 Highway Patrol managers.
She said she'd instructed both of them to come up with a reform plan for the organization.
It's not clear what has changed from last week to this, but Glover is insisting he's retiring and not resigning from the patrol. He will officially retire on Sept. 1.
In the meantime, Glover is out talking to troopers about the future of the patrol.
Glover and Young were in Monroe on Monday telling troopers to sign and accept the new ethics policy or turn in their badge.
Young defends a new policy that allows the Highway Patrol to look at a trooper's personal cell phone records even though some people are questioning its legality.
"We don't care who they're talking to, we don't care what they're talking about, we just want to make sure they're working and not spending an inordinate amount of time on the cell phone," Young said.
On Thursday, Glover and Young will visit Troop C Headquarters on Blue Ridge Road in Raleigh.