(CNN) Skepticism abounds that any measure significantly changing the nation's laws related to gun control will gain enough Republican support in the Senate to pass, coming a week after President Donald Trump's tours of Dayton and El Paso. But Trump's aides and congressional officials have begun early stage talks on some possible responses.

One such idea would expand background checks on gun sales, according to officials from both the White House and Congress, but GOP senators continue to signal to Trump that a major push for new gun laws could alienate the voters he'll need to win reelection.

According to those officials in the discussions between White House officials and staffs for Sens. Joe Manchin, Pat Toomey and Chris Murphy, the aides have ticked through the various past iterations of the background check bill sponsored by Toomey, a Republican, and Manchin, a moderate Democrat. That measure failed to gain approval in 2013, following the Sandy Hook Elementary School shooting, which took place in Murphy's home state of Connecticut.

The sessions have been "informational, not substantive," one official said. Trump's aides hope to present him a slate of options during a briefing scheduled for later this week at his Bedminster, New Jersey, golf club, where he is currently on a summer vacation. They hope to give him a realistic picture of what measures could pass in Congress, according to White House officials.

In one recent poll, voters -- Democrats and Republicans -- overwhelmingly agreed expanding background checks is a good idea . And there are signs that some rank-and-file members of Congress in competitive districts would rally behind their party's leader should Trump decide to act.

"I think we have to respond," Rep. Michael McCaul, a Texas Republican, told CNN this week. "I don't think we can put our head in the sand and allow it to happen again."

"I think anybody that buys a gun should have a background check," he added.

But there's little evidence that the recent mass shootings will compel Congress to find a solution. There have been 18 mass shootings with nine or more dead in the past 12 years, and no major gun legislation passed in that time.

Any effort faces headwinds

One challenge now confronting staffers is the number of lawmakers and their aides who are on vacation over the August recess.

And it's unclear how much pressure Republicans are feeling from their constituents back home. At a recent local Chamber of Commerce event, McCaul said he thought Congress would act, adding he wanted to make sure it did so "without infringing on our constitutional rights." He mentioned the warning signs before the Boston Marathon bombing in 2013 and said Congress could pass legislation to improve information sharing between government agencies and the monitoring of public content on the internet "to stop the event before it happens."

But the issue did not appear to be top of mind for the 30 or so people in the audience; none of the questions McCaul received were about guns.

Most of the major decisions on what deals need to be cut will have to wait until they return to Washington. The discussions have also so far also involved few Republicans, not yet accounting for the dozens of other GOP senators in deep red states who rejected the Toomey-Manchin bill in 2013.

Trump looking to take a concrete step

Still, in conversations with advisers over the weekend as he began his week-long working vacation in New Jersey, Trump has said he believes he needs to take a concrete step on gun control that is meaningful, rather than symbolic, according to people familiar with the matter. He's relayed his views to Senate Majority Leader Mitch McConnell in phone conversations over the past week, though McConnell hasn't yet endorsed any specific piece of gun legislation.

"I am convinced that Mitch wants to do something," Trump told reporters on Tuesday. "He wants to do background checks, and I do too, and I think a lot of Republicans do."

Trump has downplayed pushing for an assault weapons ban, which he doesn't believe would pass muster in the Senate. And he's said he doesn't want to expand background checks to private transfers between family members, a key point of contention in the gun debate.

"There is nobody more pro-Second Amendment than Donald Trump, but I don't want guns in the hands of a lunatic or a maniac," Trump said Tuesday. "And I think if we do proper background checks, we can prevent that."

He's been encouraged by some aides, including daughter Ivanka, to press on background checks. But others -- including another of his children, Donald Trump Jr., according to sources -- have appeared more skeptical. The President's son is an avid hunter and a key channel between the President and the conservative base who ardently opposes any attempt to restrict gun sales.

Expanding background checks has always been a heavy lift. The last major effort to pass the Toomey-Manchin bill in 2013 did not gain enough Republican support to overcome a filibuster. Aides refer often to the fact that if nothing happened after the Sandy Hook shooting, there is no reason to believe it would happen now.

Red flag bills also struggle to find support

Several conservative allies and Republican lawmakers have privately voiced opposition to his push for background checks, claiming they wouldn't have stopped the shootings in Dayton and El Paso. These officials have tried to relay their concerns to Trump, encouraging him to advocate for so-called "red flag" laws, which would allow guns to be removed from people deemed a risk to themselves or others, instead, but he has not been receptive, they told CNN.

Meanwhile, others close to the President have questioned how long his interest in and commitment to passing background checks will last. In the past -- including after the Parkland, Florida, high school shooting -- Trump has voiced strong support for strengthening checks on firearm purchases. But over time that stance softened as Republicans and the National Rifle Association made their opposition known.

There isn't evidence yet that Trump is wielding an aggressive arm-twisting campaign for a specific piece of legislation as the Senate continues its extended vacation. And Democrats are skeptical the talks or other negotiations will lead to an outcome.

"I don't feel like they are any more serious than the last 10 conversations on guns," one Democratic aide said.

Even giving states incentives to pass red-flag laws are coming up against real opposition from conservatives in the GOP conference. When Republican Sen. Lindsey Graham announced earlier this month that he reached a deal with Democratic Sen. Richard Blumenthal to give states incentives to pass some version of red flag laws, it did not sit well conservatives who have strong federalist views.

Sen. Mike Rounds, a Republican from South Dakota who has worked across the aisle on immigration and other hot-button issues, said Tuesday he hadn't seen a single red flag proposal he could support. And he downplayed the prospects that new gun proposals would meet his and other Republicans' demands.

"The key is meaningful, and we haven't seen that," he told reporters, adding later: "There's a reason why a lot of this stuff has not been done already, and that's because it's not as easy as what it sounds like."

The bill worries some voters as well.

After the local Chamber of Commerce event McCaul attended, Denise Davis, an insurance agent and gun owner, said she's worried about red flag proposals that give someone "carte blanche to make a decision without due process."

Davis added that she wanted Congress to think before it acts on any potential gun legislation.

"I don't want it to be a knee-jerk reaction where they say, 'I've got to do something because people are hollering at us to do something,'" she said.