Senate Minority Leader Charles SchumerCharles (Chuck) Ellis SchumerTurf war derails bipartisan push on surprise medical bills Senate confirms Trump's nominee to lead FDA CEO group pushes Trump, Congress on paid family, medical leave MORE (D-N.Y.) is calling on President TrumpDonald John TrumpRepublicans aim to avoid war with White House over impeachment strategy New York Times editorial board calls for Trump's impeachment Trump rips Michigan Rep. Dingell after Fox News appearance: 'Really pathetic!' MORE to tell Majority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellRepublicans aim to avoid war with White House over impeachment strategy New York Times editorial board calls for Trump's impeachment CNN's Cuomo promotes 'Dirty Donald' hashtag, hits GOP for 'loyalty oath' to Trump 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.

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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 MurphyRepublicans aim to avoid war with White House over impeachment strategy On The Money: Trump, China announce 'Phase One' trade deal | Supreme Court takes up fight over Trump financial records | House panel schedules hearing, vote on new NAFTA deal Schumer: Trump 'sold out' on China trade deal 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.

Sen. Maggie HassanMargaret (Maggie) HassanTurf war derails bipartisan push on surprise medical bills Hillicon Valley: Twitter to start verifying 2020 primary candidates | FTC reportedly weighs injunction over Facebook apps | Bill would give DHS cyber unit subpoena powers | FCC moves to designate 988 as suicide-prevention hotline Senate bill would give DHS cyber agency subpoena powers MORE (D-N.H.) said McConnell should "convene the Senate immediately for an emergency session." Meanwhile, Sen. Mazie HironoMazie Keiko HironoHorowitz offers troubling picture of FBI's Trump campaign probe Live coverage: DOJ inspector general testifies on Capitol Hill Democrats rip Barr over IG statement: 'Mouthpiece' for Trump MORE (D-Hawaii) said it was "past time" for McConnell to act.

"Call us back for a vote. Now," she added.