Supporters of "concealed-carry reciprocity" argue they simply want to clarify a patchwork of laws from local jurisdictions that confuse and sometimes ensnare gun-carrying travelers.
And they insist any bill sent to President Trump would not put state laws under a federal umbrella law or ease background checks.
"This vote marks a watershed moment for Second Amendment rights," said Chris W. Cox of the National Rifle Association's legislative affairs group.
The bill passed in the House 231-198 last week, with six House Democrats voting in support.
But Texas GOP Sen. John Cornyn, who is sponsoring the companion Senate measure, will need more than that in his chamber.
In 2013, Cornyn got support for a similar bill from 13 Democrats, including seven who are still in the Senate. Five of the seven are seeking reelection next year -- Joe Donnelly in Indiana, Martin Heinrich in New Mexico, Heidi Heitkamp in North Dakota, Joe Manchin in West Virginia and Jon Tester in Montana. All those states are conservative-leaning except for New Mexico.