WASHINGTON—Support grew among congressional Republicans Thursday for tightening restrictions on a rifle accessory that was used to terrifying effect in the Las Vegas mass shooting, raising the prospect of a rare bipartisan agreement on gun legislation.

The National Rifle Association on Thursday signaled it could support some restrictions on the devices, called bump stocks, easing the way for action since Republicans often follow the group’s lead on gun policy. The NRA and GOP lawmakers also left open the possibility that they...