Expanding background checks to include gun show sales got 84 percent support in an Associated Press-GfK poll earlier this year. Near-universal background checks have received similar or stronger support in other national polls.
Polls in some Southern states have been comparable. March surveys by the Quinnipiac University Polling Institute found more than 9 in 10 people in Florida and Virginia backing expanded background checks, the same margin found by an Elon University Poll in North Carolina in February.
Analysts say people support more background checks because they consider it an extension of the existing system. That doesn't translate to unvarnished support from lawmakers, in part because the small but vocal minorities who oppose broader background checks and other gun restrictions tend to be driven voters that politicians are reluctant to alienate.
"It's probably true that intense, single-issue gun voters have been more likely to turn out than folks who want common-sense gun laws," said Mark Glaze, director of Mayors Against Illegal Guns, the group that Bloomberg helps lead. Glaze, however, said he believes that voters favoring gun restrictions have become more motivated since Newtown and other recent mass shootings.
Several moderate Democrats are holding back as they assess the political landscape. They're also waiting to see exactly what the Senate will consider.
Sen. Mark Begich, D-Alaska, said Wednesday his state's voters tell him, "Don't take away our rights, our individual rights, our guns." Begich said he opposes a strict proposal requiring background checks for nearly all gun sales but will wait to see whether there is a bipartisan compromise he can support.
The problems faced by gun control supporters go beyond the challenge of winning over moderate Democrats. GOP opponents are sure to force Democrats to get 60 of the Senate's 100 votes to win, and there are only 53 Democrats plus two independents who generally support them.
Also targeted by Bloomberg's ads are 10 Republicans, including Sen. Jeff Flake of Arizona, home of ex-Rep. Gabrielle Giffords, who was severely wounded in a mass shooting; the retiring Sen. Saxby Chambliss of Georgia; and moderate Sen. Susan Collins of Maine.
In another indicator of hurdles facing gun control forces, the Senate voted 50-49 last week to require 60 votes for any legislation narrowing gun rights. The proposal lost because 60 votes in favor were required, but six Democrats voted for the proposal, offered by conservative Sen. Mike Lee, R-Utah.
"It confirms there's no such thing as an easy gun vote," said Jim Kessler, a senior vice president of the centrist Democratic group Third Way.
The gun bill also increases penalties for illegal gun sales and slightly boosts aid for school safety.
More abrupt changes like an assault weapons ban generally get slight majorities in polls. Democratic leaders decided to omit it from the Senate bill because such a provision lacks enough votes.