WASHINGTON (Reuters) - U.S. Democrats sought to gain the upper hand in the gun debate with President Donald Trump and Senate Republicans on Tuesday by demanding quick action on background checks and suggesting that money for Trump’s border wall could go instead to fighting domestic terrorism.

Five Democrats in the House of Representatives held a news conference with relatives and friends of mass shooting victims to pressure Senate majority leader Mitch McConnell to call the upper chamber back from its August recess for a vote on gun control legislation that has already been approved by the House.

In a related move, Senate Democratic leader Chuck Schumer was preparing to ask the Trump administration in the coming days to withdraw its request for $5 billion for a border wall with Mexico and direct the funds to initiatives to combat white supremacy beliefs and gun violence, according to a person familiar with his thinking.

Schumer wants the money to go to FBI domestic terrorism investigations, Department of Homeland Security initiatives to counter violent extremism, and federal health agency research on gun violence, the source said.

The Democratic push on gun control comes after back-to-back mass shootings in El Paso, Texas, and Dayton, Ohio, propelled the wide availability of guns in the United States back to the forefront of political debate. Democrats said that more than 250 mass shootings have occurred in the United States this year.

According to the Gun Violence Archive, a non-profit research and data collection organization, there have been 257 mass shootings so far this year in the United States. The group defines a mass shooting as an incident in which four or more people are shot or killed, not including the shooter.

With opinion polls showing a majority of Americans in favor of gun restrictions but Trump and his fellow Republicans unlikely to give in on tough measures, Democrats in Congress and those vying for their party’s presidential nomination are making it a key plank in their 2020 election contests.

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Congress has failed to pass meaningful gun control measures despite multiple mass shootings over the years, mostly due to lobbying by the National Rifle Association and other pro-gun groups. Those groups have tapped into a deep-seated suspicion of government efforts to interfere with gun ownership, which is protected by the Second Amendment of the U.S. Constitution.

Democrats are currently pressing for Senate action on a House bill known as H.R. 8 that would expand background checks by closing a loophole that allows as many as one-fifth of gun sales over the internet and at gun shows to go unchecked. The measure passed the Democratic-led House in February, getting only eight Republican votes.

“Is it the solution to all this violence? It is not ... it is a significant and important step to take in the face of this violence,” House majority leader Steny Hoyer told reporters. “Not to act is a refusal to meet our responsibilities to the American people,” he added.

Hoyer said the House Judiciary Committee, which has jurisdiction over gun control, will return to Washington early to consider new “red flag” legislation that would help states keep guns away from people deemed a threat to themselves or others, and ban both assault rifles and high-capacity magazines. A committee aide said a final decision to return has not been made, however.

McConnell told a radio interviewer last week that calling the Senate back into session early would only lead to political posturing. Instead, he has instructed the chairs of three Senate committees to work on legislation that can be debated when lawmakers return from their recess on Sept. 9.

Legislation sought by McConnell would need to garner enough bipartisan support to pass the Senate and secure Trump’s signature.

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The White House, which originally threatened to veto the House legislation, is now reassessing the package, according to sources familiar with the matter.

A Senate version that would exempt from background checks all gun transfers, including sales between close family members, was also under consideration, the sources said. The House version exempts only loans and gifts between relatives.

The Senate’s bill, sponsored by Republican Senator Pat Toomey and Democratic Senator Joe Manchin, failed in 2013 in the aftermath of the murder of 20 children and six staff members at an elementary school in Newtown, Connecticut, in December 2012.

The White House is also considering red flag measures and proposals to notify states nationwide when someone is denied a gun purchase, the sources said.

The president failed to mention toughening background checks in an address last week that focused on mental illness, the internet and video games as potential causes of shootings. But he took a different tack when the issue came up again on Tuesday.

McConnell “wants to do background checks. I do too, and I think a lot of Republicans do,” Trump told reporters in New Jersey. “I don’t know frankly that the Democrats will get us there. ... We’ll see what happens.”