Senate Minority Leader Charles SchumerChuck SchumerOvernight Energy: Schumer to trigger reconciliation process Wednesday | Bipartisan bill would ban 'forever chemicals' in cosmetics | Biden admin eyes step toward Trump-era proposal for uranium reserve GOP senator: I want to make Biden a 'one-half-term president' How Biden can get the infrastructure bill through Congress MORE (D-N.Y.) is calling on President TrumpDonald TrumpKushner lands book deal, slated for release in 2022 Biden moves to undo Trump trade legacy with EU deal Progressives rave over Harrison's start at DNC MORE to tell Majority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellOn The Money: Schumer to trigger reconciliation process on Wednesday | Four states emerge as test case for cutting off jobless benefits GOP senator: I want to make Biden a 'one-half-term president' McConnell presses for 'actual consequences' in disclosure of tax data 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 MurphyBipartisan infrastructure deal takes fire from left and right Democrats mull overhaul of sweeping election bill Rising crime rejuvenates gun control debate on campaign trail 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.