But the Senate leader all but dismissed a measure from Sen. Dianne Feinstein (D-Calif.) to let the attorney general deny firearms to suspected terrorists, noting that the GOP suspects the amendment “is basically a politically-motivated effort.”

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Democrats are aiming to present that measure as an amendment to an appropriations bill on the floor this week.

But Senate GOP leaders are calling for everyone to cool their heels until lawmakers receive an official briefing about what happened in Orlando, and how Congress can prevent such a tragedy from happening again.

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The Senate meets Wednesday with FBI director James Comey and Homeland Security Secretary Jeh Johnson, who were briefing members of the House on the Orlando attack Tuesday afternoon.

“If we want to engage in a serious legislative effort we’re going to talk to the experts about what we can do to be helpful,” McConnell said.

Late last year, the Senate rejected Feinstein’s measure in the wake of the San Bernardino, Calif., shooting. Republicans opposed it, arguing the government had made too many mistakes attempting to identify suspected terrorists on its official watch lists to trust that individuals wrongly on the list wouldn’t be denied their Second Amendment rights. The Feinstein proposal failed on a 45 to 54 vote, when just one Republican — Sen. Mark Kirk of Illinois — voted to support it.

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At the time, Sen. John Cornyn (R-Texas) presented an alternative, which would have given the attorney general or relevant U.S. attorney a 72-hour period to delay the sale of a firearm or explosive to a suspected terrorist and prove to a judge there was probable cause to turn that delay into a permanent ban. It also failed, after all Democrats but Sen. Joe Donnelly (D-Ind.) and Joe Manchin (D-WVa.) voted against it.

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According to an aide, the GOP is thinking about raising Cornyn’s amendment again as an alternative to Feinstein’s should it come to a vote on the floor. But no final decisions have been made.

Other Republican senators signaled Tuesday they might be ready to consider a gun ban for suspected terrorists, though they described something that falls short of Feinstein’s measure.

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Sen. Lindsey Graham (R-S.C.) told reporters he was “willing to work with my colleagues to say that if you’re being investigated by the FBI because of your actions, that that’s enough to get you into a system” that would make it so such people “can’t buy a gun that day.”

“I just want to make sure that if you’re on by mistake, there’s some due process for the person trying to buy a gun,” Graham said.

Democrats argue there would be ample recourse for anyone mistakenly barred from procuring a gun under Feinstein’s proposal, as people prohibited from obtaining a firearm could file suit against the government if necessary.

Mike DeBonis contributed to this report.