Senate Minority Leader Charles SchumerCharles (Chuck) Ellis SchumerTurkey says soldier killed despite cease-fire in Syria Schumer calls for FDA to probe reports of contaminated baby food How Trump and Pelosi went from bad to worse MORE (D-N.Y.) is calling on President TrumpDonald John TrumpTrump says he doesn't want NYT in the White House Veterans group backs lawsuits to halt Trump's use of military funding for border wall Schiff punches back after GOP censure resolution fails MORE to tell Majority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellTrump urges GOP to fight for him Senate Dems signal they'll support domestic spending package Trump's top picks for Homeland Security chief are ineligible for job: reports 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 MurphyThe Hill's Morning Report - Presented by Better Medicare Alliance - Trump has had a rough October Pelosi, Schumer hit 'flailing' Trump over 'sham ceasefire' deal Romney slams ceasefire deal, calls Trump's Syria move 'a bloodstain' in US history 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) HassanHillicon Valley: GOP lawmakers offer election security measure | FTC Dem worries government is 'captured' by Big Tech | Lawmakers condemn Apple over Hong Kong censorship Sen. Hassan calls for look into federal government support for entities hit by ransomware attacks Hillicon Valley: Senate passes bill to boost cyber help for agencies, businesses | Watchdog warns Energy Department failing to protect grid | FTC sues Match for allegedly conning users MORE (D-N.H.) said McConnell should "convene the Senate immediately for an emergency session." Meanwhile, Sen. Mazie HironoMazie Keiko HironoDemocrats introduce SWAMP Act to ban meetings with foreign leaders at Trump properties Democrats urge Rick Perry not to roll back lightbulb efficiency rules Overnight Energy: Lawmakers show irritation over withheld Interior documents | Republican offers bipartisan carbon tax bill | Scientists booted from EPA panel form new group MORE (D-Hawaii) said it was "past time" for McConnell to act.

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