The Senate will vote Thursday on whether to move ahead with a controversial gun safety bill that some critics say could open the door to a national firearms registry.

Senate Majority Leader Harry Reid, D-Nev., wouldn't promise the support of his entire Democratic caucus, but he nonetheless appeared confident there are enough votes to block a GOP-led filibuster so that debate on the bill will proceed.

"We need to move this legislation," Reid said after meeting privately with Senate Democrats on Tuesday afternoon.

Reid made the announcement as nearly a dozen family members of the 26 Newtown, Conn., shooting victims made their way around the Capitol in an effort to convince lawmakers to back tough new gun safety measures, including an assault weapons ban and limits on high-capacity magazine clips.

The families ducked media requests, but according to the Senate's two Connecticut lawmakers, they plan to push for passage of the strongest gun safety measure possible when they meet face-to-face with senators this week.

"Nobody can make the case better than theses families," Sen. Chris Murphy, D-Conn., said after meeting with them.

The bill up for consideration does not include an assault weapons ban or limit on magazine clips the families want included in the bill, though they could be added later as amendments.

Instead, it requires universal background checks for gun purchases.

The bill would also provide federal funding for enhanced school safety and would increase penalties for gun trafficking.

Even without the stronger provisions, fourteen Republicans and at least two Democrats are considering a vote to filibuster the legislation. Sens. Mark Pryor, D-Ark. and Max Baucus, D-Mont., who represent pro-gun states and face tough re-election battles in 2014, said they were unsure.

Universal background checks have gained opposition from the pro-gun lobby because many believe they could require the federal government to create a gun registry.

"My primary emphasis is on the people I work for," Baucus said Tuesday when asked whether he would attempt to block the bill. "I'm going to do what's right for the people of Montana."

Democrats control 55 votes, and without Pryor and Baucus, Reid would need at least seven Republicans to reach the 60-vote threshold needed to block a filibuster.

An informal count by The Washington Examiner indicated at least seven GOP senators are likely to vote against a filibuster and to begin debate on the bill, including Sens. John McCain of Arizona and Susan Collins of Maine.

Collins said she, too, is opposed to the universal background check provision in its current form but might support a compromise measure being written by Sens. Joe Manchin, D-W.Va., and Pat Toomey, R-Pa., that would exclude hunters and close family members from having to comply.

"I never thought it made any sense to require a background check when a father gives his daughter a gun, unless he knows she is not permitted under the law to own a firearm," Collins said. "But that is not the case with the vast majority of family transfers or gifts."

More than a dozen GOP senators have pledged in a letter to Reid that they would filibuster the gun bill, among them Ted Cruz of Texas.

Cruz said that even with language exempting families from background checks, he remains opposed to the bill.

"If a so-called universal background check were to pass, the immediate response of the Obama administration would be to say we cannot enforce it without a registry," Cruz said. "That's the next step."