(CNN) - Senate Democrats are hardening their position over new gun legislation, dismissing attempts to pass narrower measures until Senate Majority Leader Mitch McConnell schedules a vote on a sweeping background checks bill already approved by the House.
But it's still possible that Senate Democrats could ultimately agree to support those narrower plans, especially if the Senate holds a vote and ultimately rejects the background checks bill, according to multiple Democratic sources familiar with the matter. There are still several steps that have to happen first before any deal is ultimately cut -- and it's likely to take weeks to play out.
Behind the scenes, McConnell has dispatched several chairmen to begin charting the course to see what can pass their chamber and could garner President Donald Trump's support -- far short of the House-passed background checks bill or an assault weapons ban sought by Democrats.
Among the items under consideration: so-called red flag legislation; bolstering funding for programs to help people with mental health issues; and finding new ways to regulate the content of violent video games.
Sen. Lamar Alexander, chairman of the Senate Health, Education, Labor and Pensions Committee, has been dispatched to determine which programs could be strengthened to help treat those who have mental health problems, according to multiple sources. That could include identifying areas where more funding is needed, such as for community health centers and school counselors.
Asked about her preference, aides to Sen. Patty Murray, the top Democrat on the committee, signaled that her interest instead is on keeping the focus on the House-passed background checks bill.
"Sen. Murray believes the urgent step the Senate needs to take right now is voting on the bipartisan gun reforms the House has already passed," Murray spokeswoman Helen Hare said.
The one bill that seems to have the most momentum is the red flag legislation, and there are several different proposals being floated that would empower authorities to deny access to firearms if an individual is deemed a risk. Florida Sen. Marco Rubio has authored a bipartisan bill, and California Sen. Dianne Feinstein has pushed forward her version backed by several Democrats. Another one is being drafted by Senate Judiciary Chairman Lindsey Graham and Sen. Richard Blumenthal, a Connecticut Democrat.
A red flag law enables those who have seen warning signs to seek a court order to intervene and temporarily prevent someone who is in crisis from having access to a firearm. For example, under California's red flag law, it's legal for family members to ask a judge to remove firearms from a relative who appears to pose a threat.
Shortly after Trump's remarks on Monday, Trump ally Graham announced he and Blumenthal had reached an agreement on a federal grant program to assist with enforcing existing red flag laws and to "encourage" more states to adopt red flag laws.
On Wednesday, Senate Minority Leader Chuck Schumer threw cold water on the red flag bills by calling them "half-measures," saying they must be paired with background checks legislation. But he was careful not to rule those proposals out.
Without specifying, Schumer seemed to dismiss both the bipartisan proposals that Rubio and Graham have floated, saying some measures Republicans are pursuing could do "more harm than good" if federal grants provided to states under the legislation ultimately lead states to develop weak red flag laws.
"We Democrats are not going to settle for half-measures so Republicans can feel better and try to push the issue of gun violence off to the side," Schumer said, adding that his caucus "will seek to require" that any such bill must be accompanied by a vote on the House universal background checks bill.
But even if the House background checks bill gets a vote on the Senate floor, it's almost certain to fall short of the 60 votes needed to overcome a filibuster. At that point, Democrats say, alternatives proposals could be pursued.
This story has been updated.
CNN's Lauren Dezenski contributed to this report.
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