The proposal from Reps. Seth Moulton, D-Massachusetts, Carlos Curbelo, R-Florida, Scott Rigell, R-Virginia, and Bob Dold, R-Illinois, would prevent people on the federal government's no-fly and Selectee lists from buying guns, and provide a mechanism to appeal a denial.
"Simple reforms like this are demanded by the American people," Moulton said in a news conference.
But the lawmakers started working on the proposal before Democrats waged a 25-hour sit-in on the House floor for gun control votes, a tactic that exasperated Republicans and emboldened Democrats.
The House members admitted Friday that they don't have the buy-in from leadership to bring the compromise to the floor -- one day after a vote in the Senate effectively left Collins' proposal in legislative purgatory.
House Majority Leader Kevin McCarthy's office did not return a request to comment on the measure.
"We are going to be pressing the Speaker's office when we get back in session. The pressure will be continuous but respectful," said Rigell, an NRA member who owns several guns, including an AR-15.
Curbelo said it was incumbent on the members to sell the measure to colleagues -- a tall task for a highly-charged issue in an election year.
"If we do nothing, I think we have failed," Rigell said. "It will be a moral failure of this institution."