Washington (CNN) House Speaker Nancy Pelosi on Monday privately urged House Democrats to stay focused on Senate Majority Leader Mitch McConnell in their push to enact sweeping background check legislation amid an intra-party debate about whether to take a sharply different strategy in the aftermath of two mass shootings.

Among the ideas: bringing the House back to session during the August recess to pass an assault weapons ban and other bills, but Pelosi counseled Democrats on Monday that their most effective strategy is to keep the pressure focused exclusively on the Senate GOP leader's resistance to allowing a vote on a background checks bill that passed out of the Democrat-controlled House earlier this year, according to sources on a private conference call.

Top congressional Democrats also publicly renewed calls on Monday for the Senate to reconvene and pass background check legislation. Pelosi and Senate Minority Leader Chuck Schumer said, "It is incumbent upon the Senate to come back into session to pass this legislation immediately," referring to the Bipartisan Background Checks Act of 2019, which passed the House in February.

McConnell responded Monday evening with a statement saying, "Senate Republicans are prepared to do our part" and that he spoke with three committee chairmen about possible solutions.

"I asked them to reflect on the subjects the President raised within their jurisdictions and encouraged them to engage in bipartisan discussions of potential solutions to help protect our communities without infringing on Americans' constitutional rights," McConnell said.

He added later, "Partisan theatrics and campaign-trail rhetoric will only take us farther away from the progress all Americans deserve."

The Democratic congressional leaders criticized President Donald Trump in their joint statement, calling him a "prisoner to the gun lobby and the NRA" and accusing him of backing off a "call for stronger background check legislation."

On Monday morning, Trump floated the idea of tying immigration legislation to strengthening background checks, tweeting, "Republicans and Democrats must come together and get strong background checks, perhaps marrying this legislation with desperately needed immigration reform."

But in a speech later in the day, the President failed to endorse any measures that would expand background checks for firearm sales, despite his apparent endorsement of such laws earlier.

Some Democrats push for more gun control efforts

In a private conference call on Monday with House Democrats, Pelosi made clear that they need to maintain their focus on McConnell to move forward on background check legislation, according to multiple sources on the call.

But some members pushed for other measures to be taken up, including an assault weapons ban and the so-called red flag bill, which would empower authorities to ensure people who pose a threat cannot access firearms.

Red flag legislation is backed by lawmakers in both parties, and the President has signaled his support for it, though there are differences in how the measures are structured. Senate Judiciary Chairman Lindsey Graham, a South Carolina Republican, released a statement Monday saying that he had talked to Trump about red flag law grants and had "reached an agreement" with Democratic Sen. Richard Blumenthal of Connecticut "to create a federal grant program to assist and encourage states to adopt 'Red Flag' Protection Order laws."

Nevertheless, Pelosi claimed to lawmakers that the families she has spoken with who have been victims of gun violence want the focus to be trained on McConnell and the background check legislation because they believe it would save the most lives, according to one of the sources on the call. Rep. James Clyburn, a South Carolina Democrat, agreed, arguing that moving other measures could undercut efforts to get the Senate to act on background checks.

The speaker said that no legislation is off the table, saying that they needed to be timed accordingly and that the House Democrats need to keep the focus on McConnell.

Sources told CNN that Democratic Rep. Veronica Escobar -- who represents El Paso -- raised the possibility of the House returning to session in August. She said on the call that she'd be willing to come back Tuesday if they're called into session, but her spokesman says her focus right now is the community.

Multiple sources on the call said there was discussion about just the House Judiciary Committee coming back in August -- rather than the full House -- to advance some gun safety bills in the mix, like red flag legislation and a ban on high-capacity magazines.

Two sources on the call said that Democrats debated whether to reconvene the House in August, but Pelosi and Clyburn didn't embrace the idea. The fear is that bringing the House back would take the focus off McConnell, sources said.

Judiciary Chairman Jerry Nadler, a New York Democrat, raised the prospect of moving an assault weapons ban, according a source on the call. Rep. David Cicilline, the Rhode Island Democrat who introduced the bill, also pushed for bringing it up, according to a different source on the call.

Calls from Democrats grow louder for Senate action

There is so far no indication, however, that the Senate majority leader would move to take up the background check legislation.

The Senate and House are in a summer recess period and are not scheduled to be in session during the month of August in Washington.

McConnell put out statements responding to the mass shootings in El Paso, Texas, and Dayton, Ohio, over the weekend. On Sunday, McConnell tweeted, "Sickening to learn this morning of another mass murder in Dayton, Ohio overnight. Two horrifying acts of violence in less than 24 hours. We stand with law enforcement as they continue working to keep Americans safe and bring justice."

Separately, the Senate majority leader's office announced on Sunday that McConnell had fractured his shoulder and is now recovering while working from his home in Louisville, Kentucky.

For now, Democrats don't appear poised to end their calls.

The joint statement from Pelosi and Schumer comes after a number of high-profile congressional Democrats, including 2020 presidential contenders, argued over the weekend that McConnell should bring the Senate back into session for the purpose of passing background check legislation.

In a Monday speech, the President called on the nation to condemn racism and white supremacy, but stopped short of acknowledging his own divisive and racist rhetoric.

Trump did not mention specific measures to limit access to firearms except proposing "red flag" laws. He also did not tie any gun action to immigration.

This story has been updated with additional developments Monday.