Senate Democrats are zeroing in on a push to require universal background checks for gun purchases following last week's shooting at a Florida high school.

Senate Minority Leader Charles Schumer Chuck SchumerVideo of Lindsey Graham arguing against nominating a Supreme Court justice in an election year goes viral Graham signals support for confirming a Supreme Court nominee this year Pelosi orders Capitol flags at half-staff to honor Ginsburg MORE (D-N.Y.) said Wednesday that universal background checks would be the party's "No. 1 priority."

"Our No. 1 priority is going to be universal background checks, which is supported by about 80 percent of the American people, and closing the gun show loophole and all the other ways that people get around the background checks," he told reporters.

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Lawmakers have turned their focus to the background check system after last week's shooting at a high school in Parkland, Fla., where 17 people were killed.

Sen. Dick Durbin Richard (Dick) Joseph DurbinMcConnell focuses on confirming judicial nominees with COVID-19 talks stalled Senate Republicans signal openness to working with Biden Top GOP senator calls for Biden to release list of possible Supreme Court picks MORE (D-Ill.) said Congress "must act by closing loopholes in gun laws and passing universal background checks" while Sen. Richard Blumenthal (D-Conn.) added that President Trump Donald John TrumpUS reimposes UN sanctions on Iran amid increasing tensions Jeff Flake: Republicans 'should hold the same position' on SCOTUS vacancy as 2016 Trump supporters chant 'Fill that seat' at North Carolina rally MORE needs to "support universal background checks, bans on assault weapons and high-capacity magazines."

Trump tweeted Tuesday evening that "we must now focus on strengthening Background Checks." And the White House said this week the president supports a bill by Sens. John Cornyn John CornynCalls grow for Biden to expand election map in final sprint Bipartisan praise pours in after Ginsburg's death Chamber of Commerce endorses McSally for reelection MORE (R-Texas) and 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 (D-Conn.) aimed at strengthening the National Instant Background Check System (NICS).

But Schumer said Wednesday that was a "small step" and "not close to enough."

"In terms of getting something real done, the universal background check is at the, sort of, nexus of [having] a chance of actually becoming law, particularly if the president would support it, and at the same time doing a whole lot of good," he said.

The House passed its version of the Cornyn-Murphy bill last year, but attached it to legislation allowing people to use permits for carrying concealed handguns across state lines.

But linking the two issues is largely considered a nonstarter in the Senate, where Republicans will need the support of at least nine Democrats to get any bill through the chamber.

The Cornyn-Murphy bill does not include the concealed carry reciprocity provision. Schumer said Wednesday that it would be a "very, very bad idea" to try to merge the two proposals into one bill.

"That would be a repeat of what they tried to do, what the administration tried to do, on immigration. Attach something good ... to something so totally unacceptable," he said.