Senators from both parties signaled Sunday that universal background checks could be the next gun measure to get shoved toward the legislative scrap heap or significantly watered down.

Sen. Charles E. Schumer, whose bill to expand background checks to cover virtually all gun transactions, including private sales, has cleared the Judiciary Committee, said he is nevertheless hunting for a compromise on the issue.

“I’m working very hard with both Democrats and Republicans, pro-NRA and anti-NRA people, to come up with a background check bill that will be acceptable to 60 senators and be very strong and get the job done,” the New York Democrat said on NBC’s “Meet the Press.” “It’s very hard. We’re working hard, and I’m very hopeful that we can get this passed.”

Two Republican senators who are backing legislation to clarify the definition of people adjudicated mentally ill who are not allowed to own or purchase guns appeared skeptical that Mr. Schumer’s measure could pass the Senate as written.

Sen. Jeff Flake of Arizona called universal checks “a bridge too far for most of us.”

“Well, the paperwork requirements alone would be significant, and even if there are exemptions for a father passing on a gun to his son or daughter, you’d still have issues with people in a private setting transferring or loaning a gun for somebody,” Mr. Flake said later on the program.

Sen. Lindsey Graham of South Carolina said that out of about 80,000 people who failed background checks last year, just 66 were prosecuted.

“Why in the world would you expand that system if you’re not enforcing the law that exists today to include private transfers?” Mr. Graham said on CNN’s “State of the Union.” “So I think that legislation is going nowhere. But I would like to have a robust debate about improving the system to make sure that people who are mentally ill do not get a gun to begin with. And there’s a lot we could do in a bipartisan fashion.”

A month after 20 children and six adults were shot to death at Sandy Hook Elementary School in Newtown, Conn., President Obama rolled out a package of gun control proposals that included bans on so-called assault weapons and high-capacity magazines as well as near-universal background checks on all gun sales.

Senate Majority Leader Harry Reid, Nevada Democrat, opted to leave the weapons and magazine bans out of the “base bill” he plans to bring to the floor after Congress’ two-week Easter break. He said the weapons ban could not get the 60 votes needed to overcome a potential filibuster and get the entire package to the floor.

Mr. Reid did say that he would ensure senators the opportunity to introduce the bans as amendments.

“In order to be effective, any bill that passes the Senate must include background checks,” he said.

Bipartisan talks involving Mr. Schumer and several others reached an impasse over the issue of record keeping on gun sales, so Mr. Schumer went ahead with his own bill.

Sen. Richard Blumenthal, Connecticut Democrat and a member of the Senate Judiciary Committee that passed Mr. Schumer’s measure on a party-line vote, said Sunday on CNN’s “State of the Union” that there is “room for compromise” on the issue. Currently, all sales by licensed firearms dealers must go through background checks, but transactions between private individuals do not.

Lawmakers are looking for a way to extend checks to almost all transactions without creating a record-keeping system that gun rights supporters fear could turn into a gun registry.

Host Candy Crowley asked whether a gun bill could be considered a success if the proposed weapons and high-capacity magazine bans are rejected and the background check bill is watered down.

“Any step that saves lives is a step in the right direction,” said Mr. Blumenthal. “And the question is not winning or losing here, but really saving lives, which the people of Newtown and the victims there and their families, I think, want to happen, not just for the sake of those victims, but also more than 3,000 people have perished since Newtown as a result of gun violence.”

But Mark Kelly, a retired astronaut and husband of former Rep. Gabrielle Giffords, said any bill that does not include a universal background check is “a mistake.”His wife was wounded in a shooting spree in Tucson, Ariz., in January 2011.

“When you use words like gun control, you know, gun control doesn’t poll very well, but we do know that over 90 percent of Americans support a universal background check,” Mr. Kelly said on “Fox News Sunday.” “And there is incredible momentum in Congress and around the nation to get this done.”

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