Begich declined to directly state his position and said of Alaskans, "We like our guns."
There are 53 Senate Democrats and two independents who lean Democratic.
In a written statement, Sen. Mike Lee, R-Utah, said conservatives want to prevent Obama from rushing the legislation through Congress "because he knows that as Americans begin to find out what is in the bill, they will oppose it."
Senate Minority Leader Mitch McConnell, R-Ky., has said he will join the conservatives' attempt to block debate.
The bill would expand required federal background checks to nearly all gun transactions — a provision Reid would try to replace with the completed agreement between Manchin and Toomey. It would also stiffen penalties for illegal firearms trafficking and provide a small boost in school safety aid.
Eleven Sandy Hook family members representing eight of the shooting victims were on Capitol Hill to lobby senators from both parties for gun legislation, including Isakson.
"We bring a very personal perspective," said Mark Barden, who lost his 7-year-old son, Daniel. "People should listen to what we have to say and move the debate forward. It's not just about our tragedy. Lots of kids are killed every day in this nation. We have to help lead the change."
Some relatives had breakfast with Vice President Joe Biden at his residence in the Naval Observatory. Later, Biden spoke to law enforcement officials at the White House and told reporters that conservatives would not succeed in blocking debate.
"This is not one of these votes that they block a vote and somehow we're going to go away," Biden said. "The American public will not stand for it."
The president's gun-control proposals have hit opposition from the National Rifle Association, which was using the Internet and emails to urge its members — it claims nearly 5 million — to tell members of Congress of their opposition.
In GOP-heavy Louisiana, where Democratic Sen. Mary Landrieu is seeking re-election next year, the NRA wrote to its members, "Please contact Senator Landrieu and encourage her to oppose this anti-freedom legislation."
Counteracting that has been an effort by Mayors Against Illegal Guns, one of whose leaders is billionaire New York City Mayor Michael Bloomberg.
The group was running a television ad in Pennsylvania asking voters to contact Toomey and tell him to back expanded background checks. It also said it will keep track of key gun-related congressional roll calls and make the information available to voters and contributors — a tactic long used by the NRA and other groups.
Associated Press writers Nedra Pickler, Jim Abrams, Andrew Miga and Henry C. Jackson contributed to this report.