Such a change would likely delay consideration of the measure until at least the end of the week, The Huffington Post reported.
Sens. Pat Toomey, R-Pa., and Joe Manchin, D-W.Va., fear they are on track to fall short of the 60-vote supermajority needed to overcome conservative opposition to their measure, The New York Times and Post said.
So the lawmakers hope making changes to their amendment -- such as exempting gun sellers located far from a federal firearms licensee from having to perform background checks on gun buyers -- may increase support, the Times and Post said.
The revised plan would seek to win over rural senators who fear the proposed background checks would be overly burdensome for gun sellers in far-off places, the Times and Post reported.
Manchin and Toomey hope Sen. Lisa Murkowski, R-Alaska, will put forward the change, the Post said.
Her office had no immediate comment.
The changes would likely delay Senate consideration of the background-check bill to Friday or even next week from Wednesday, the Post said.
The Toomey-Manchin amendment, as currently written, would close loopholes for gun buyers who have been able to avoid background checks by buying their firearms online or at gun shows.
Of the 16 Republicans who joined 50 Democrats and two independents in voting last week to consider gun legislation, more than a half-dozen said they would not support the Toomey-Manchin amendment, and other GOP lawmakers said they had not made up their mind.
Manchin and Toomey touted their compromise on the Senate floor Monday, appealing for bipartisanship and saying their measure had many provisions to strengthen gun rights.
"If you're a believer in the Second Amendment right to bear arms, you're going to like this bill," Manchin said.
At the same time, Sen. Tom Coburn, R-Okla., said he intended to introduce his own background-check measure. It would mandate universal background checks through the Internet, similar to Toomey-Manchin, but would not require sellers to keep a record of the transaction -- something law enforcement says is critical to tracking illegal guns.
"The proposal I will offer would create a consumer portal that would allow someone to go online for free and print out a pass that proves they are not on the [National Instant Criminal Background Check System] list," Coburn said in a letter to colleagues.
"Law-abiding citizens won't be treated as guilty until proven innocent and they won't face a new tax as they exercise their constitutional rights," his letter said. "Citizens also won't be required to keep records under my proposal.
"Finally, my bill will allow people who already have a concealed carry permit to buy a gun without taking additional steps, and it will give states the right to come up with their own ways to declare that someone isn't on the NICS list," he said in his letter.
Coburn plans to promote his amendment as bipartisan, even though it has no Democratic co-sponsor, because he believes it will get broad, bipartisan support if the Manchin-Toomey approach fails, the Times said.
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