NEW YORK — (CNN) A Navy official tried to throw more cold water Wednesday on armed private citizens showing up on their own at military recruitment centers.

The official told CNN that that should unauthorized armed civilians attempt to patrol outside a Navy recruiting center, recruitment personnel would try to work out of the office. They could choose to spend the day at a school, mall, on travel or other duties that would keep them away from the center.

The official noted, however, that so far only one or two of these armed civilians have shown up following the shooting at a Chattanooga, Tennessee, recruiting center Thursday that left five servicemen dead.

Earlier Wednesday, Department of Defense spokesman Capt. Jeff Davis reiterated the existing department policy of not supporting arming all military personnel in comments to reporters.

Lawmakers and elected officials at the national, state and local levels have proposed changing the policy following the deadly shooting Thursday.

“We do not support arming all military personnel,” Davis said.

On Tuesday, the Marine Corps put out a statement saying, “While we greatly appreciate the support of the American public during this tragedy, we ask that citizens do not stand guard at our recruiting offices. Our continued public trust lies among our trained first responders for the safety of the communities where we live and work.”

Chairman of the Joint Chiefs of Staff Gen. Martin Dempsey is expected to be briefed by Thursday by the services and senior Defense officials on their recommendations for improving security at military installations, according to two Pentagon officials.

Dempsey will then consolidate all the data and present a single set of recommendations to Defense Secretary Ashton Carter by Friday, the deadline the secretary imposed on the Pentagon. The officials say the recommendations are not expected to be made public anytime soon.

The-CNN-Wire ™ & © 2015 Cable News Network, Inc., a Time Warner Company. All rights reserved. (Photo: CNN)