Sen. Pat Toomey, R-Pa., said he and Sen. Joe Manchin, D-W.Va., had until probably Wednesday to secure the 60-vote supermajority needed to overcome conservative opposition to the measure, which would close loopholes for gun buyers who have been able to avoid background checks by buying their firearms online or at gun shows.
Sixteen Republicans voted last week to begin debate on the legislation, but aides say fewer GOP lawmakers are expected to support the bipartisan amendment.
"I think it's an open question as to whether or not we have the votes," Toomey said Sunday on CNN's "State of the Union." "I think it's going to be close."
"What we're asking for ... is just for our colleagues to read it," said Manchin, who appeared with Toomey on the program.
Manchin said he believed the measure -- a centerpiece of broad gun legislation -- would get the 60 votes needed to pass after his colleagues read it.
Manchin -- who, like Toomey, is a gun owner -- has already met with several Senate colleagues about the background-checks proposal, including with Sen. John Thune, R-S.D., in the Senate gymnasium, Manchin staff members told The Wall Street Journal.
Thune's office had no immediate comment.
Sen. Susan Collins, R-Maine, who Manchin buttonholed during a lunch last week, told NBC News Saturday she planned to support the proposal.
Manchin additionally hand wrote cover notes to about 25 senators of both parties and attached them to a fact sheet about his amendment, an aide told the newspaper.
The amendment offered by Toomey and Manchin would replace the background-check provision in legislation about to hit the Senate floor.
It would be the first amendment to the bill, which also has provisions to create tougher penalties for the so-called straw purchasing of guns, when people buy firearms for other who are ineligible, then lying if asked about the identity of the gun's ultimate possessor.
Other proposed amendments, including a ban on certain semiautomatic rifles known as "assault weapons" and a limit to high-capacity ammunition magazines, are expected to fall short when they come up for votes this month.
Toomey and Manchin won conditional backing from Sen. John McCain, R-Ariz., who said on the CNN program he would likely support the measure.
"I'm very favorably disposed toward that," he said. "Eighty percent of the American people want to see a better background-check procedure."
The National Rifle Association, the nation's largest gun-rights group, opposes the background-check amendment, calling it misguided and ineffective in preventing further violence.
The NRA's lobbying arm told lawmakers last week it would consider their vote on the amendment in its rankings of legislators.
Manchin told CNN if the NRA doesn't support the bill, "we just agree to disagree.
"But there's things in this piece of legislation that they've been working for many, many years to get, and it's here," he said.
He called NRA officials "my friends," adding, "My door is still open."
But in a break from the usually united gun-rights lobbying stand, the Citizens Committee for the Right to Keep and Bear Arms, the No. 2 gun-rights lobbying group, said Sunday it would support the Toomey-Manchin amendment.
Manchin told Fox News Channel he considered the endorsement "huge."
CCRKBA Chairman Alan Gottlieb said the Toomey-Manchin measure expands "a good number of pro-gun rights measures," such as strengthening veterans' ability to purchase firearms and lifting current restrictions on interstate handgun sales.
Independent Firearm Owners Association President Richard Feldman told the Journal his group conditionally backed the measure but wanted to review the amendment's language before formally endorsing it.
He said the group supported gun-show background checks because "you're selling to people you don't know who they are."
Gun Owners of America, another gun-rights lobbying group, sent out an alert asking its members to pressure their lawmakers not to support the amendment, the group's legislative counsel, Michael Hammond, told the Journal.