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Mr. Graham responded by saying one thing everyone should be able to agree on is that the background check system for gun purchasers and the laws already on the books should work properly.

“I guess the point is if we don’t want the wrong people to own guns, which we all agree, the one way to do that is to take the system that’s supposed to make the distinction between a person who should and shouldn’t and enforce it,” he said. “I own an AR-15. I passed a background check. Isn’t it really about who has the gun, sometimes more than the gun itself?”

But across town, speaking in front of a group of attorneys general from across the country, Mr. Biden issued a warning to those who want to oppose gun controls, saying voters in Chicago delivered a message Tuesday. Robin Kelly, a staunch gun control advocate, defeated a host of other candidates, including one who has received backing from the National Rifle Association, to become the Democratic nominee in a special election for the seat vacated by disgraced former Congressman Jesse Jackson Jr.

“The message is: there will be a moral price as well as a political price to be paid for inaction,” Mr. Biden said.

Nevertheless, any legislation that makes it through the Senate would still have to clear the Republican-controlled House. Rep. Bob Goodlatte, Chairman of the House Judiciary Committee, poured a bit of cold water on Mr. Biden’s proclamation, saying he didn’t think universal background checks — to say nothing of more ambitious proposals such as bans on so-called assault weapons and high-capacity magazines — would be a part of the House’s work on the issue.

“I think where we’re going to find the ability to produce legislation is going to be focused on two things,” the Virginia Republican said at a breakfast hosted by the Christian Science Monitor. “One is improving the background check system, but universal background checks I do not think will be part of that. The other one will be improving the efforts to crack down on illegal sales of firearms on the streets, if you will — people who knowingly engage in transactions where they engage in selling firearms to people who should not be able to purchase them.”

Stephen Dinan contributed to this report.