(Reuters) - The governors of Florida, Texas and Indiana on Saturday ordered National Guard members at offices and other facilities to be armed in the wake of attacks that left five servicemen dead in Tennessee.

Governors Rick Scott of Florida, Greg Abbott of Texas and Mike Pence of Indiana said National Guard members need to protect themselves after four Marines and a Navy petty officer were shot and killed on Thursday in Chattanooga, Tennessee. One site was a recruiting office in a strip mall.

In Florida, six guard offices now in storefronts will move into armories.

"It has become clear that our military personnel must have the ability to defend themselves against these type of attacks on our own soil," Abbott said in a news release.

"Arming the National Guard at these bases will not only serve as a deterrent to anyone wishing to do harm to our service men and women, but will enable them to protect those living and working on the base," he added.

In Florida, Scott said Guard members who do not carry weapons should get them and obtain expedited concealed weapon permits, if necessary. Pence ordered the arming of military personnel at all Indiana National Guard facilities and recruiting storefronts.

Military personnel in uniform would not usually have authorization to carry a personal weapon while working in recruiting offices, the Pentagon said on Friday.

Scott ordered six storefront Florida National Guard recruiting centers to be moved to the nearest Guard armory buildings.

In his executive order, Scott said bulletproof glass and better video surveillance equipment are among steps that should be considered to make recruiting offices safer.

(Reporting by Kevin Murphy in Kansas City; Editing by Alan Crosby)