Senators will vote Monday on gun control proposals after Democrats waged a near 15-hour filibuster to force a debate on the issue.
But it would also let the attorney general delay the sale of a gun to anyone who has been the subject of a terror investigation within the past five years.
Cornyn told repoters earlier Thursday that he was optimistic he would be able to pick up more Democratic support next week, because he isn't including a provision that would have cracked down on cities that don't support federal immigration laws.
The Senate will also take votes on competing background check proposals.
A measure from GOP Sen. Chuck GrassleyChuck GrassleyBig Tech critics launch new project Senate antitrust bill has serious ramifications for consumers and small businesses Hillicon Valley: Amazon's Alabama union fight — take two MORE (Iowa) would reauthorize and provide funding for the National Instant Background Check System (NICS), provide incentives to share mental health records and and bolster federal record sharing.
Grassley and Sen. Ted CruzRafael (Ted) Edward CruzSupreme Court appears divided over Cruz campaign finance challenge Democrats, poised for filibuster defeat, pick at old wounds O'Rourke says he raised record .2M since launching campaign for Texas governor MORE (R-Texas) previously introduced a background check bill in 2013 during the Senate's debate on the Manchin-Toomey proposal, but their measure failed by a 52-48 vote.
The Senate will also take a procedural vote on a proposal from Democratic Sens. Chris MurphyChristopher (Chris) Scott MurphyBiden huddles with group of senators on Ukraine-Russia tensions Overnight Defense & National Security — Texas hostage situation rattles nation Senators to meet with Ukraine president to reaffirm US support MORE (Conn.), Chuck SchumerChuck SchumerDemocrats make final plea for voting rights ahead of filibuster showdown DACA highlights pitfalls of legalization schemes The Hill's Morning Report - Presented by Facebook - Schumer tees up doomed election reform vote MORE (N.Y.), Richard Blumenthal (Conn.) and Cory Booker (N.J.) to expand background checks.
Their proposal would require — with a handful of narrow exceptions — a background check for the sale and transfer of any gun.
It would also impose penalties for states that don't make data for NICS electronically available and would require federal agencies to certify that they have handed over all records on any individual that would be prohibited from buying a gun.
Because of how McConnell filed cloture on the four proposals the senators will vote on the background check proposals, before moving to the fight over suspected terrorists being able to buy a gun.
—Updated at 5:56 p.m.