Personal transfers between family members, for example, would be exempt from background checks.
The bipartisan initiative spearheaded by Sens. Joe Manchin (D-West Virginia) and Pat Toomey (R-Pennsylvania) is also trying to strengthen the existing background check system by encouraging all states to contribute to it.
Finally, it hopes to establish a national commission on mass violence to study causes of mass shootings like the one in Newtown, Conn., last December, Manchin's website says.
The legislation could face an early vote on Thursday, Yahoo News said.
“The events of Newtown changed us all,” Manchin said at a press conference today, Yahoo reported. “Nobody here … with a good conscience could sit by and not try to prevent a day like that from happening again.”
On his website, Toomey said the Public Safety and Second Amendment Rights Protection Act will not take away guns, ban any types of guns, restrict ammunition or create a firearms registry.
Quickly, the National Rifle Association said it opposed the legislation.
“Expanding background checks at gun shows will not prevent the next shooting, will not solve violent crime and will not keep our kids safe in schools,” the NRA posted on its website.