Senate Minority Leader Charles SchumerCharles (Chuck) Ellis SchumerJohnson eyes Irish border in Brexit negotiations Lewandowski on potential NH Senate run: If I run, 'I'm going to win' Appropriators warn White House against clawing back foreign aid MORE (D-N.Y.) is calling on President TrumpDonald John TrumpFacebook releases audit on conservative bias claims Harry Reid: 'Decriminalizing border crossings is not something that should be at the top of the list' Recessions happen when presidents overlook key problems MORE to tell Majority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellPelosi, Schumer press for gun screenings as Trump inches away The malware election: Returning to paper ballots only way to prevent hacking First House Republican backs bill banning assault weapons MORE (R-Ky.) to bring up a stand-alone background check bill, dismissing Trump's suggestion of tying it to immigration reform.

"Instead of flailing around blaming everything under the sun, if the president is serious about 'strong background checks' there’s one thing he can do: Demand Sen. McConnell put the bipartisan, House-passed universal background checks bill up for a vote," Schumer tweeted on Monday.

Trump said on Monday that he wants legislation providing "strong background checks," including potentially linking it to long-stalled immigration reform. Background check legislation already faces an uphill climb in the GOP-controlled Senate, but tying it together with immigration legislation would all but guarantee it would not get passed.

Talk of new background check legislation comes after two mass shootings rocked the nation over the weekend.

A gunman on Saturday killed 20 people and wounded dozens more in an attack at a Walmart in El Paso, Texas, on Saturday. Less than a day later, at least nine people were killed and more than two dozen were injured in a Dayton, Ohio shooting. The two shootings are not believed to be linked.

Democrats are trying to build pressure on McConnell to bring up background check legislation after the two shootings, including calls for the GOP leader to bring the Senate into session during the August recess.

The Senate left for the five-week August recess on Thursday and are out of session until Sept. 9. A spokesman for Schumer didn't immediately respond to a question about if Democrats will try to clear a bill by unanimous consent during pro forma sessions this week. The request would likely be blocked by a GOP senator.

Sen. Chris MurphyChristopher (Chris) Scott MurphyOvernight Defense: US, Russia tensions grow over nuclear arms | Highlights from Esper's Asia trip | Trump strikes neutral tone on Hong Kong protests | General orders ethics review of special forces White House eyes September action plan for gun proposals Trump phoned Democratic senator to talk gun control MORE (D-Conn.) echoed Schumer's demand during a tweet on Monday morning saying Trump asking McConnell to bring up a bill would ensure its passage.

"Background checks has already passed the House - w Republican votes. If Trump asked McConnell to support it, it would pass in a week. FYI - he won’t do that," he tweeted.