Texas is poised to become the largest state in the U.S. to allow citizens to openly carry handguns, a change long sought by gun-rights activists.

The Texas House of Representatives on Friday voted 96-35 to allow residents with concealed-handgun licenses to openly carry their guns in public in holsters. A similar open-carry measure passed the Texas Senate last month; the two open-carry bills must be squared before being sent to Republican Gov. Greg Abbott, who has indicated support for the idea.

In contrast to its reputation for being permissive on firearms, Texas is one of six states, including California, New York and Florida, that currently bars citizens from openly carrying handguns. People who want to carry handguns in public on their person must obtain concealed-weapons permits and keep the weapons hidden.

“We are seeing historic progress in Texas”- Terry Holcomb Sr., executive director of Texas Carry, a gun-rights group

Texas currently allows citizens to openly carry long guns in public, however. That has spurred gun-rights groups to carry assault rifles into restaurants and stores and along the sidewalks adjoining the Texas Capitol, to highlight what they see as a senseless legal distinction.

The push by gun enthusiasts, a powerful constituency within the Texas Republican Party, has helped legislation move through the GOP-controlled Legislature this year after several years of stalled efforts.

Gun-rights groups are also calling for legislation to allow some students and university employees to carry concealed handguns on college campuses. The Texas Senate last month approved a campus-carry measure, as it is known. The Texas House is expected to pass a companion measure, and the idea is also supported in principle by Mr. Abbott.

It would make Texas one of only eight states with laws permitting concealed guns on college campuses, and the largest state to do so.

“We are seeing historic progress in Texas,” said Terry Holcomb Sr., executive director of Texas Carry, a gun-rights group. He noted that open-carry legislation had never even made it out of a Texas legislative committee before this year.