WASHINGTON — On the heels of two mass shootings that have shocked the country, the Senate is, once again, weighing an overhaul of the nation's gun laws — with a murky path ahead. Despite President Joe Biden's pleas on Tuesday, two House-approved bills to close gaps in the background checks system don't currently have the 60 votes needed to clear the Senate, according to senators who have spoken with colleagues to gauge support. Democrats say they're determined to act, seeing it as a necessity to get some measures across the finish line while they control the White House and Congress. But getting 10 Republican senators to agree on any form of gun control, however popular or modest, will be tough. "I think a universal background checks bill can get 60 votes," said Sen. Chris Murphy, D-Conn., an outspoken advocate for stricter gun control. March 23, 202101:58 "We're going to sit down with Democratic leadership this week and talk about the path forward," he said. "I think we've got two weeks of recess in which I think there'll be a lot of conversations, across the aisle, about the path forward on background checks." Asked about gun legislation Tuesday, Senate Minority Leader Mitch McConnell said he's "open to the discussion" but didn't say what ideas he might support. The Kentucky Republican said he opposes the House-approved policies, adding, "What I'm not attracted to is something that doesn't work." Without 60 votes in the Senate, gun control measures are likely to fuel calls by some in the Democratic Party to nix the filibuster and allow measures to be passed with just 50 votes. Democratic Sens. Joe Manchin of West Virginia and Kyrsten Sinema of Arizona say they don’t support abolishing the filibuster. And should activists call for pairing gun control laws with getting rid of the filibuster, it’s unclear that alone could produce enough pressure to get the Senate to change its rules. The more aggressive gun control proposals that Biden pushed on Tuesday, to "ban assault weapons and high-capacity magazines," are less likely to get even 50 votes in the Senate. Murphy noted the popularity of extending background checks, and argued that its failure would call into question the viability of the super-majority requirement to pass legislation in the Senate. "If a measure that has 90 percent to 95 percent public support can't pass the Senate just because of our rules — not because it doesn't get the majority of support in the Senate — then something's really wrong here," Murphy said. "Democracy dies when things that have the majority of support in Congress, the support of the president and 90 percent public support can't become a law." Manchin-Toomey Discussions are likely to center on bipartisan legislation proposed in 2013 by Manchin and Sen. Pat Toomey, R-Pa., that would extend federal background check rules to transfers by unlicensed sellers at venues like gun shows and on the internet, which currently occur without checks in some states. That proposal, known as Manchin-Toomey, couldn't get 60 votes in the Senate at the time. Manchin told NBC News on Tuesday he has "no idea" how many votes the bill could get today. His partner in the effort said it would be an uphill climb. "I still support background checks on commercial sales," Toomey said. "We're having preliminary conversations and I hope we can get something across the goal line. But, you know, it’s very difficult." March 23, 202103:49 Toomey declined to offer more details about conversations he is having or say who he is talking to, but did acknowledge he doesn't think the House-passed background check bills can pass the Senate. That bill, known as H.R. 8, is stricter than the Manchin-Toomey proposal and requires FBI background checks for private gun sales, including transferring guns to family members, with limited exemptions for "a loan or bona fide gift" between certain family members. Sen. Susan Collins, R-Maine, who supported Manchin-Toomey in 2013, called the current House gun checks bill "very, very broad." Other Republicans said there's hope for action. "Tightening the background check system is possible. The House bill is too broad and goes too far for the Senate," said Sen. Rob Portman of Ohio. "But I think something is possible." Manchin, notably, said he doesn't support the House-passed background check bill, but still favors his 2013 proposal, calling it reasonable, responsible and a matter of "gun sense." "I come from a gun culture," Manchin told reporters. "Commercial transactions should be background checked. Commercial — you don't know a person. If I know a person, no." 'It tears at your heart' This isn't the first time that shootings have grabbed the nation's attention and focused Congress on gun laws. But it's the first time in more than 10 years that Democrats have controlled both chambers of Congress and the White House. The Senate Judiciary Committee held a hearing Tuesday on gun laws, in which the chairman, Sen. Dick Durbin, D-Ill., called for a "moment of action." Senate Majority Leader Chuck Schumer, D-N.Y., promised Tuesday he'd put gun measures to a vote in the chamber, including the House-approved bills. On Monday, a shooting at a grocery store in Boulder, Colorado, left 10 people dead, including one police officer. Last week, shootings at three Atlanta-area spas left eight people dead. "We've had so many mass shootings, it tears at your heart. My heart goes out to the people there, and people across our country who are suffering," Sen. Mitt Romney, R-Utah, told reporters on Tuesday. Asked what legislation can pass, Romney replied, "That's a big topic." Republicans are likely to keep pushing for state-focused solutions. Sen. Marco Rubio of Florida wouldn't say Tuesday if he'd vote for the House-passed background checks bills, and instead proposed creating a federal fund that encourages states to establish "red flag" laws like one enacted in Florida. That measure allows family members or police to keep guns away from a person if they can convince a judge that the person is a threat to himself or herself, or to others. "And so our bill creates a fund that incentivizes more states to pass those kinds of laws," Rubio said. "I haven't even seen the House bills, but I'm not here to focus on bills that won't prevent any of these crimes."