Virginia Gov. Bob McDonnell did not outwardly endorse such a plan, but cautioned against immediately rejecting it.
"I know there's been a knee jerk reaction against that," McDonnell said to the idea when asked in an interview on WTOP, a Washington radio station. "I think there should be a discussion of that - if people were armed, not just the police officer but other school officials were trained and chose to have a weapon, certainly there would have been an opportunity to stop aggressors coming in to the schools."
His comments follow an endorsement of the idea by Gov. Rick Perry, R-Texas, Monday night. He said teachers with the proper training and license should have "access to weapons in their school" and licensed Texans "should be able to carry your handgun anywhere in this state."
McDonnell pointed to armed airline pilots as an example of non-law enforcement professionals carrying firearms on the job. Thousands of pilots have gone through training and been authorized to carry a firearm aboard their airplane.
The state's current law bans possession of a firearm by anyone other than law enforcement within 1,000 feet of a school -- a measure McDonnell said he supported.
Before this term as governor, McDonnell served as attorney general of his state from 2006-2009, which included the 2007 killing of 33 people at Virginia Tech.
He spoke about watching news of the Friday shooting unfold on his television, and said it is time for "another discussion about mental health and mental public safety and how those intersect," and time to consider "how are we going to be a more just and fair and compassionate society so people don't act out like this."
"My hope is that we're not at that point where when a little first or second grader comes in to school, the first thing they see is a police officer with a gun," he said in the interview
Arming school personnel in the most recent school shooting could have saved lives, McDonnell suggested. The gunman who forced his way into Sandy Hook Elementary School killed 20 students and six adults in a barrage of bullets, police said, before taking his own life.
"This perpetrator went into the principal's office and actually killed the principal that was lunging, according to the facts that have come out so far, at the perpetrator heroically to try to stop him," he said. "If a person was armed and trained could they have stopped the carnage in the classroom? Perhaps. I think these are discussions that rational people should have to see what works to be able to stop school violence."