Washington (CNN) As President Donald Trump weighs legislative and executive options to address gun violence in the US, his own instinct to expand background checks is butting up against intense pressure from gun rights advocates, who are pressuring him that such a move would harm him politically.

When Trump delivered prepared remarks to a nation mourning two mass shootings last weekend, he proposed "red flag" laws , which enable family members and authorities to get court orders to keep potentially dangerous relatives from having access to firearms, but did not mention specific measures to more broadly limit access to firearms.

Since then, however,Trump has said expanding background checks makes sense and that he believes most people agree with him. A Quinnipiac poll from March found 73% of registered voters thought "more needs to be done to address gun violence" in the United States, including majorities across parties (95% of Democrats, 74% of independents and 52% of Republicans). That poll also found 86% in support of the background check bill passed by the House, described as requiring "background checks on all gun sales, including those at gun shows and through online retailers."

Trump on Friday morning indicated support for tightening background checks, but also noted his close ties to the National Rifle Association, which has come out against expanding background check systems.

"Serious discussions are taking place between House and Senate leadership on meaningful Background Checks. I have also been speaking to the NRA, and others, so that their very strong views can be fully represented and respected," Trump tweeted. "Guns should not be placed in the hands of mentally ill or deranged people. I am the biggest Second Amendment person there is, but we all must work together for the good and safety of our Country. Common sense things can be done that are good for everyone!"

Potential executive action

The Trump administration has touted a few victories to restrict guns -- banning bump stocks, such as the one used by a shooter who killed 58 people in Las Vegas in 2017, and signing the Fix NICS Act into law in 2018, which aims to improve government agencies' and states' reporting to the National Instant Criminal Background Check System.

But it's unclear what background check provisions the President would support now.

Despite the administration's previous indications that the President has been open to expanding background checks, Trump has backed down . And in February, he threatened to veto two pieces of gun control legislation: one bill that would extend the FBI's review period for an incomplete federal background check from three business days to 10 and another that would impose a universal background check nationwide. Both bills have passed the House but have not been voted on in the Senate.

A source familiar with internal discussions says Trump is looking at ways to expand background checks through some sort of executive action. But the source cautions it's early in the process.

And the White House is now refusing to stand by the veto threats from earlier this year.

White House press secretary Stephanie Grisham said Thursday, when asked about the veto threat, that "the administration is looking into many options."

Trump speaks with NRA chief after mass shootings

LaPierre stressed to Trump that they don't think the calls for more restrictive gun measures in Washington match how his supporters in deep red areas feel about the issue.

LaPierre issued a statement Thursday night saying in part, "I'm not inclined to discuss private conversations with President Trump or other key leaders on this issue."

He added, "But I can confirm that the NRA opposes any legislation that unfairly infringes upon the rights of law-abiding citizens. The inconvenient truth is this: the proposals being discussed by many would not have prevented the horrific tragedies in El Paso and Dayton. Worse, they would make millions of law abiding Americans less safe and less able to defend themselves and their loved ones."

The NRA also still opposes the pair of background check bills passed by the House earlier this year and continues to have reservations about a federal "red flag" law.

"To safeguard the rights of law-abiding gun owners, state Extreme Risk Protection Orders at a minimum must include strong due process protections, require treatment, and include penalties against those who make frivolous claims," the gun rights group said of red flag laws in a statement.

In addition, the group said the "so-called 'universal' background check bill passed by the House would not have prevented the criminal acts in El Paso or Dayton."

A person familiar with the discussions said the NRA harbors deep reservations about red flag laws due to the potential for abuse -- that people could have their guns stripped away by a court with few due process protections. The gun rights group has laid out a number of conditions for supporting such laws -- including demanding that a judge be required to determine whether a person who has been "red-flagged" meets the state standard for involuntary commitment -- a high bar.

But aides on Capitol Hill are questioning the strength of the NRA.

Aides say the reality is the NRA is not as strong as it once was. Organizational infighting has led to questions about whether the organization will have the influence it had in the next election. And while the membership is still strong, expanding background checks generally polls highly even among gun owners.

Asked generally about the NRA dynamic, one GOP Senate aide said, "This might be a good moment to do something."

"There is a growing disdain for that part of the gun culture that thinks the essence of the Second Amendment is being able to walk around with a military-grade weapon," the aide said.

Gun lobbyists, meanwhile, don't appear pleased with the solutions pitched so far.

A source familiar with the gun lobby's thinking on the solutions being proposed echoed the concerns about red flag laws and due process and said none of the ideas floated so far are workable.

It's "the same issue with universal background checks," the sources said, added that they create a sense of "false security."

These efforts are "trying to stop fish from going through a net," the source continued.

McConnell weighs options

It's unclear which bills Senate Majority Leader Mitch McConnell, a Kentucky Republican who singularly holds power to take up legislation in the Senate, is willing to bring to the floor.

McConnell told a Kentucky radio station Thursday the Senate will put the issue of background check legislation in addition to red flag laws "front and center" when the body reconvenes after its summer recess, but it will not return early as Democrats are demanding. McConnell made no commitments on what would come to the Senate floor, suggesting that the negotiations are ongoing.

McConnell had said Monday evening, in response to the shootings , that "Senate Republicans are prepared to do our part."

The top two congressional Democrats each spoke with Trump on Thursday about the issue, according to a news release, after House Speaker Nancy Pelosi urged the President in a letter to call the Senate back to Washington to deal with the House-passed legislation.

The California Democrat said in a joint statement with Senate Minority Leader Chuck Schumer, a New York Democrat, "We spoke to the President separately this afternoon and told him the best way forward to address gun violence in our country is for Leader McConnell to let the Senate take up and pass the House-passed universal background checks legislation and for the President to sign it into law. The President gave us his assurances that he would review the bipartisan House-passed legislation and understood our interest in moving as quickly as possible to help save lives."

Behind the scenes, aides say McConnell has dispatched three chairmen to begin working on proposals that could attract bipartisan support and be signed by the President, proposals that in his words "help protect our communities without infringing on Americans' constitutional rights." Aides close to the process say the lawmakers are looking at potential mental health legislation, the role of video games and ways to give states incentives to enact red flag laws.

And Vice President Mike Pence said Thursday that the White House has spoken to McConnell about red flag laws as Republicans weigh next steps.

An aide also said Sen. Lindsey Graham, by virtue of his chairmanship of the Judiciary Committee, will be tasked with just how far to go on background checks. The South Carolina Republican and Democratic Sen. Richard Blumenthal of Connecticut are working on a proposal that would create a federal grant program that encourages states to impose their own red flag laws, which exist now in only 17 states plus Washington, DC.

But the Senate aide said that anything could be possible if Trump put his sustained weight behind it.

Republican Sen. Pat Toomey of Pennsylvania -- a key proponent of expanding background checks on gun sales -- said he was encouraged by the President saying there was a "great appetite" to address the issue.

Toomey and the President have spoken "several" times this week about the long-stalled background check bill he co-sponsored with Democratic Sen. Joe Manchin of West Virginia, an aide to Toomey said, adding that Trump has "expressed a willingness to work with Sens. Toomey and Manchin on the issue."

Toomey, however, hedged on whether the President would give his full support to the bill, telling CNN's Jake Tapper on Tuesday that the President seemed "open" to supporting it. And Manchin told reporters Wednesday that he still does not know what gun proposals the President would potentially sign on to, despite having at least two conversations with Trump this week.