Republican gubernatorial candidate Ken Cuccinelli II signaled Wednesday that he was open to letting trained teachers carry guns in schools, while Democratic rival Terry McAuliffe called for an armed officer at every elementary campus that wants one.

Their comments followed a rampage through a Connecticut school, where a gunman killed 20 first-graders and six staff members before taking his own life.

They also came a day after Gov. Robert F. McDonnell (R) remarked in a radio interview that Virginia should consider arming teachers, principals and other school staff to protect children from such attacks — an idea some local education leaders sharply criticized Wednesday.

Cuccinelli, the state’s attorney general, said through a spokesman that he was “open to the discussion” about allowing teachers and other staff to carry weapons.

“[A]s the governor indicated, examining other options — such as arming school resource officers and other properly trained personnel who choose to carry firearms — should be part of any discussion in a review of safety procedures at Virginia schools,” said Cuccinelli spokesman Brian J. Gottstein.

“The AG is open to the discussion,” Gottstein added. “Who ultimately fits the description of ‘properly trained personnel who choose to carry firearms’ would be considered in the school safety review process.”

McAuliffe spokesman Brennan Bilberry said it made more sense to deploy professional officers than to arm teachers.

“Terry’s mainstream proposal to extend the option of School Resource Officers to elementary schools is a better step we can take to protect Virginia children,” Bilberry said via e-mail. “These officers are already professionally trained to handle dangerous situations and would have the protection of children as their primary responsibility. Terry’s idea is a proven, common sense solution.”

Lt. Gov. Bill Bolling (R), who dropped out of the race for the GOP gubernatorial nomination in November but says he is mulling an independent bid, broke with McDonnell by dismissing the possibility of arming teachers.

“The Lieutenant Governor believes that the job of a teacher is to teach and he does not support arming teachers,” Bolling’s deputy chief of staff, Ibbie Hedrick, said via e-mail. “If school security needs to be enhanced, it should be done by trained law enforcement personnel.

Del. Robert G. Marshall (R-Prince William) is proposing a bill that would require schools to arm some teachers or other staff. That goes beyond McDonnell’s comments; the governor had said there should be a discussion about whether staff should be allowed to carry concealed weapons.

The idea of arming teachers was not warmly received by Northern Virginia superintendents, one of whom called the concept “absurd.”

“I’m going to be very simple. I’m going to be very straightforward,” said Pat Murphy, superintendent of Arlington Public Schools. “Absolutely not.”

Fairfax County Public Schools Superintendent Jack Dale said he hadn’t seen the details of any proposals but added: “the concept of having educators armed in the classroom, to me, is absurd. If you want trained law-enforcement officers in schools you should hire trained law-enforcement officers for the schools. It’s nonsensical to ask educators to carry weapons.”

Dale said he had received half a dozen e-mails from parents Wednesday, “pleading with me, ‘Don’t you dare entertain a proposal to arm faculty in the schools.’ ”

And in support of the idea? “Zero. None whatsoever.”

The issue hasn’t come up locally in Prince William County Public Schools, said spokesman Phil Kavits, but he said the school system welcomes the opportunity to join in statewide discussions about school safety. A spokesman for Loudoun County Public Schools declined to comment.

“There are a lot of issues we need to explore in the aftermath of this tragedy — but the solutions are not going to come from increasing access to weapons,” said Meg Gruber, president of the Virginia Education Association, in a statement Wednesday. “That’s a step in the wrong direction. I’ve not heard from a single teacher or administrator who said that they want to go to school armed with a gun.”

McAuliffe, former chairman of the Democratic National Committee, called for legislation that would help localities provide an officer to any elementary school that requests one.

“We already have these trained officers at the vast majority of high schools and middle schools in Virginia and should simply extend this common sense option to as many elementary schools as possible,” he said. “These officers are professionals who are specifically trained to respond to dangerous situations and would be available to deal with any threat our children might face.”

McAuliffe, who also praised steps taken by the term-limited McDonnell to review school safety across the state, said the cost of such a program would be “manageable” and partly covered by federal funds.

“[I]t is of course the decision of each school whether to pursue this option,” he said. . . . “I will be encouraging Delegates and Senators from both parties to support mainstream legislation to this effect that would improve the safety of our children with limited cost to the Commonwealth.”

McAuliffe has also called for a ban on assault weapons.

Gottstein said Cuccinelli is primarily focused on mental-health issues related to such shootings.

As a lawyer in private practice, Cuccinelli served as a court-appointed attorney for people in Virginia’s involuntary civil commitment process, Gottstein noted. As a state senator, Cuccinelli pushed for easing the involuntary commitment process, limiting gun ownership rights for the mentally ill and breaking down barriers to the sharing of information between doctors and our court system, Gottstein said.

“The most pressing issue Attorney General Cuccinelli is focused on is looking at how mental illness has played a major factor in so many of these tragedies, and that is also where he has tremendous expertise,” Gottstein said.