According to a copy of the bill text provided to ABC, it would go into effect 180 days after its passage.
“It shall be unlawful for any person to import, sell, manufacture, transfer or possess, in or affecting interstate or foreign commerce, a trigger crank, a bump-fire device or any part, combination of parts, component, device, attachment or accessory that is designed or functions to accelerate the rate of fire of a semiautomatic rifle but not convert the semiautomatic rifle into a machine gun,” the bill states.
The ban would not apply to sales or possession of the devices by U.S. agencies or departments.
Republican leaders have pushed back on Democrats' calls for tightening gun control legislation after the shooting, calling the demands premature.
"I think it's particularly inappropriate to politicize an event like this," Senate Majority Leader Mitch McConnell, R-Ky., said Tuesday. "The investigation's not even been completed. And I think it's premature to be discussing legislative solutions if there are any."
"In the meantime, our priority is on tax reform, as my colleagues have indicated," McConnell added.
ABC News' Benjamin Siegel contributed to this report.