Some potential amendments could broaden gun rights and weaken supporters' backing for the overall bill.
One proposal is by Sens. Lindsey Graham, R-S.C., and Mark Begich, D-Alaska, who say it would improve how the federal background check system blocks weapons from going to people with certain mental problems, though critics say it would make it harder in some cases to do so. Another possible amendment would require states to recognize permits for carrying concealed weapons issued by other states.
In Thursday's vote, 50 Democrats and 2 Democratic-leaning independents were joined by 16 Republicans in voting to begin debate on the legislation. Twenty-nine Republicans and two Democrats facing re-election next year in GOP-leaning states voted "no" -- Alaska's Begich and Mark Pryor of Arkansas.
After the roll call, Obama spoke by phone with some Newtown families and said he would "keep fighting for the votes they deserve," said White House spokesman Jay Carney.
The next significant vote is expected next week, when Reid tries adding an amendment expanding background checks less broadly than the legislation being debated. Broadening the system to cover more transactions is the heart of the current effort on guns.
That amendment, a compromise between Sens. Patrick Toomey, R-Pa., and Joe Manchin, D-W.Va., would subject buyers in commercial settings like gun shows and the Internet to the checks but exempt non-commercial transactions like sales between friends and relatives.
That accord, unveiled Wednesday, was designed to build bipartisan support for the legislation and seemed likely to do so. Toomey and Manchin are among the most conservative members of their parties and are both gun owners with NRA ratings of "A."
Toomey said Thursday he believes supporters of his compromise with Manchin would be able to beat back any filibuster attempt.
"Beyond that, I just don't know yet," he said on "CBS This Morning."
Gun-control groups have embraced the Manchin-Toomey compromise with varying degrees of enthusiasm, and they continued to applaud it Thursday -- while also expressing concerns about some provisions.
Besides the exemption for private sales, gun control advocates expressed displeasure with language letting gun dealers sell handguns to out-of-state customers, exempting some holders of permits for concealed weapons from background checks and shielding individuals who sell guns from some negligence lawsuits.
"We are optimistic that this bill will make a dramatic difference in reducing gun violence," said Dan Gross, president of the Brady Campaign to Prevent Gun Violence.
The NRA said it opposed the Manchin-Toomey compromise and warned senators that it would count votes on provisions it opposes in its evaluation of candidates that it provides its members, who the organization says number nearly 5 million.Associated Press writers Erica Werner, Charles Babington and Laurie Kellman contributed to this report.