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Senate Democratic Leader Harry Reid is teeing up a Thursday test vote on sweeping new gun regulations, as supporters ratchet up efforts to pressure wavering lawmakers in the face of mounting roadblocks from conservatives.

Democratic leaders are hoping to build a bipartisan coalition strong enough to overcome Republican threats to filibuster, which would require 60 votes to proceed. But Reid acknowledged Thursday he's not sure if Democrats have enough support to advance the legislation.

"If they don't help me invoke cloture on this bill, we're going to vote on these things anyway," Reid said Tuesday. "It will take a little bit of time but as I've said for months now, the American people deserve a vote on background checks, on federal trafficking, on safety in school, on the size of clips, and on, yes, assault weapons."

Cloture is the Washington word for overcoming a filibuster and, in this case, proceeding to debate on the bill.

At least 14 Republican senators have threatened to filibuster the bill, complaining it would curb the rights of the law-abiding while doing little to reduce crime. "The more people learn about the consequences of current gun control proposals, the less they will support the new restrictions," Sen. Mike Lee, R-Utah, said.

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If Lee and his colleagues follow through, Reid would need at least a handful of Republicans on board for Thursday's test vote. Fox News confirms that at least seven Republicans have said they would not join the filibuster. While there are 55 members in the Democratic caucus, Reid could lose some moderate Democrats -- meaning Thursday's vote will be close. Even if Reid overcomes that hurdle, it's unclear whether he could again marshal that support to approve the legislation.

The lobbying was well underway Tuesday on Capitol Hill to try and build such a coalition. As negotiations continued on a compromise measure that could sway moderate hold-outs, powerful advocacy groups were rolling out new ads and campaigns to win them over.

Family members of the victims of the Sandy Hook mass shooting are in Washington through Thursday to meet with senators on both sides of the aisle and lobby for legislation. After flying back from an event with President Obama on Air Force One Monday night, the family members held a conference call Tuesday afternoon. They are calling for universal background checks, limits on high-capacity magazines and tougher gun trafficking laws.

Separately, New York Mayor Michael Bloomberg's group Mayors Against Illegal Guns launched a scoring system Tuesday to give lawmakers a letter grade on their gun control stances. This is similar to what the National Rifle Association does in scoring lawmakers' gun-rights positions.

"For too long, the only voice that has been loud enough to influence Congress has been the Washington gun lobby's -- that's how we've ended up with ineffective gun laws that have fueled our country's gun violence epidemic," Bloomberg said in a statement. "Now we're working to make sure that the voices of the more than 900 bipartisan mayors in our coalition -- and the 90 percent of Americans who support commonsense reforms like background checks for all gun sales -- are heard loud and clear."

Obama's chief campaign arm Organizing for Action on Monday also began running online ads on Facebook and search engines asking Americans to urge senators to support universal background checks.

These groups are targeting moderate lawmakers seen as pliable on the issue of gun control. The new efforts come after the legislation appeared to be losing momentum. First, Reid dropped a renewed ban on assault weapons and high-capacity magazines from the main bill -- though they could still get a vote as an amendment. Then, senators began to clash over the issue of universal background checks.

On the sidelines, West Virginia Democratic Sen. Joe Manchin and Pennsylvania Republican Sen. Pat Toomey are now discussing a possible compromise proposal on the checks, which could serve to assuage gun-rights supporters' concerns about the new system encumbering casual transactions among family members and fellow hunters.

Late Monday, Senate Republican Leader Mitch McConnell joined conservatives threatening to filibuster and demand a 60-vote threshold.

"Sen. McConnell opposes the Reid bill (S.649). While nobody knows yet what Sen. Reid's plan is for the gun bill, if he chooses to file cloture on the motion to proceed to the Reid bill, Sen. McConnell will oppose cloture on proceeding to that bill," spokesman Mike Brumas said.

Obama criticized the proposed filibuster during a campaign-style event at the University of Hartford in Connecticut about 45 miles from the elementary school where 20 first-graders were shot and killed in December.

"Some back in Washington are already floating the idea that they might use political stunts," said Obama, who was introduced by Nicole Hockley, whose son, Dylan, was one of the victims. "They're not just saying they'll vote 'no' on ideas that almost all Americans support. They're saying they won't allow any votes on them at all."

Many in the crowd responded by chanting: "We want a vote."