House Speaker John Boehner, R-Ohio, repeated his view that the Senate should act, saying, "It's one thing for two members to come to some agreement. It doesn't substitute the will for the other 98 members."
In a written statement, Obama said he'd prefer stronger language than the compromise, but he said it represented progress.
"It recognizes that there are good people on both sides of this issue, and we don't have to agree on everything to know that we've got to do something to stem the tide of gun violence," he said.
Other highlights of Obama's gun agenda — including bans on assault weapons and high-capacity ammunition magazines — seem to have little chance of winning approval in the Senate, let alone the House.
Polls show more than 8 in 10 people back expanded background checks. Even so, the fight will be difficult in both chambers, especially the House, where increasing numbers of district lines are drawn to protect incumbents, said James Pasco, executive director of the Fraternal Order of Police.
"They're not going to have a constituency in every instance that is champing for that bill," said Pasco, whose group has backed the drive for expanded background checks.
The director of Mayors Against Illegal Guns, one of whose leaders is New York City Mayor Michael Bloomberg, said the compromise was a major improvement and showed that Republican support was possible.
"Opposing yet another reform, which has Americans scratching their heads, is not a place where a national political party can afford to be," said Mark Glaze.
Reflecting concerns about unseen details about some gun rights language, the Brady Campaign to Prevent Gun Violence took a more tempered view.
"While we continue to review the legislation, we believe a majority of the components are a good step forward," said Brian Malte, director of the campaign's network mobilization.
There are no current, definitive statistics on how many gun sales occur annually and what portion occurs without the checks. A study in the 1990s found that up to 40 percent of transactions involved no checks.
Associated Press writers Jim Abrams, Andrew Miga and Kenneth Thomas contributed to this report.
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