Sen. Lindsey Graham Lindsey Olin GrahamLincoln Project mocks Lindsey Graham's fundraising lag with Sarah McLachlan-themed video The Hill's Morning Report - Sponsored by Facebook - Republicans lawmakers rebuke Trump on election Trump dumbfounds GOP with latest unforced error MORE (R-S.C.) said President Trump Donald John TrumpFederal prosecutor speaks out, says Barr 'has brought shame' on Justice Dept. Former Pence aide: White House staffers discussed Trump refusing to leave office Progressive group buys domain name of Trump's No. 1 Supreme Court pick MORE is looking for a way to expand background checks for firearms sales, which could lay the groundwork for a bipartisan deal after a string of mass shootings this summer.

Graham, a top ally of Trump's who has lead jurisdiction over gun control measures as chairman of the Senate Judiciary Committee, expressed optimism about a potential measure after spending time with Trump the previous day.

“The White House is working with Sen. Murphy from Connecticut. I talked to Sen. Murphy. We’ve got some bumps but I think we’re getting there in a space where we can expand background checks to cover more commercial transactions,” Graham said of negotiations between Trump and Sen. Chris Murphy Christopher (Chris) Scott MurphyDemocratic senator calls for 'more flexible' medical supply chain to counter pandemics The Hill's 12:30 Report - Presented by Facebook - Don't expect a government check anytime soon GOP chairman to release interim report on Biden probe 'in about a week' MORE (Conn.), a leading Democrat on the issue of gun violence.

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“Most people feel like I do, you know, giving a gun to your son or daughter is not going to trigger a federal background check, but we can do other things,” Graham added.

“We’re trying to expand background checks,” he said.

Trump has sent mixed signals on where he is on the issue of expanding background checks for gun sales following mass shootings this summer in Ohio and Texas that killed dozens.

Under current law, sales made through federally licensed firearms dealers are subject to background checks, but not sales between individuals who are not dealers.

Immediately after the mass shootings in Dayton, Ohio, and El Paso, Texas, early last month, Trump tweeted that “serious discussions” were taking place between Senate and House leaders on “meaningful background checks.”

But early this month the president appeared to back away from strengthening background checks, telling reporters that doing so likely wouldn’t have prevented recent mass shootings.

He instead spoke more generically about “strong measures to keep weapons out of the hands of dangerous and deranged individuals” and reforms to the nation’s mental health system.

In late August, Trump also said the nation already has “very, very strong background checks” for firearms sales.

Senate Republican Whip John Thune John Randolph ThuneHouse to vote on resolution affirming peaceful transition of power Trump dumbfounds GOP with latest unforced error Senate passes resolution reaffirming commitment to peaceful transition of power MORE (S.D.), however, said that Trump is still weighing the option of strengthening background checks and did not rule out legislation sponsored by Sens. Pat Toomey Patrick (Pat) Joseph ToomeyAppeals court rules NSA's bulk phone data collection illegal Dunford withdraws from consideration to chair coronavirus oversight panel GOP senators push for quick, partial reopening of economy MORE (R-Pa.) and Joe Manchin Joseph (Joe) ManchinThe debate over the filibuster entirely misses the point Trump plans to pick Amy Coney Barrett to replace Ginsburg on court Day before Trump refused to commit to peaceful transition, Aaron Sorkin described how he would write election night MORE (D-W.Va.) at a White House meeting Tuesday afternoon.

“I wouldn’t say that they’re ruling things out. I think they’re trying to figure out what is achievable. So there’s still a lot of discussion about that,” he said.

“They’re looking at a lot of different ideas that individual senators who have bills have proposed or members who don’t have bills but have ideas are proposing and trying to distill that.”

Thune added that the president and his advisers are trying to “figure out if there’s a path forward that would get us a result.”

“There are some ideas out there that, to me, perhaps could actually pass,” he said.