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Under rules adopted this year, Mr. Reid could force a bill onto the floor in exchange for allowing at least two amendments to Republicans and two amendments to members of his own party. But using that method would be a signal that the bill is unlikely to survive a filibuster later in the process.

Because of that, Democrats prefer to have a final compromise bill ready to go before they begin to vote. They were hoping to begin floor action this week, but with a compromise elusive, action could be pushed back.

Like Mr. Reid, Mr. Obama on Monday blasted Republicans for considering a filibuster. He also seemed to reply to some in his own party who said he needs to create a “sense of urgency” for stronger gun controls.

“You know what? This isn’t about me,” Mr. Obama said. “This is not about politics. This is about doing the right thing for all the families who are here that have been torn apart by gun violence.”

The White House will continue its effort to pressure Congress for action with events this week, including a meeting of the vice president and attorney general with law enforcement Tuesday at the White House, and first lady Michelle Obama traveling to Chicago on Wednesday to talk about gun violence.

Republicans argue that the debate isn’t about politics but about fundamental freedoms enshrined in the Second Amendment. They also question whether restrictions on types of guns and magazines will work to keep firearms out of the hands of criminals.

“Senators have an obligation both to uphold the Constitution and to promote solutions that effectively address national problems,” Sen. Mike Lee, Utah Republican, said Monday. “The current gun control proposals fail both tests.”

As Mr. Obama was trying to rally support in Connecticut, some in Washington were holding out hope for a deal.

A spokesman for Sen. Tom Coburn said the Oklahoma Republican is trying to reach a bipartisan deal on background checks. He had been negotiating with Mr. Manchin and Sens. Charles E. Schumer, New York Democrat, and Mark Kirk, Illinois Republican.

Sen. Chuck Grassley of Iowa, the ranking Republican on the Senate Judiciary Committee, also is considering alternate measures that could include improvements to the FBI’s National Instant Criminal Background Check System, providing resources to help address mental health problems and school safety, protection for veterans from false health determinations, and gun trafficking.

He may approach those items as separate issues or in a package, depending on what Mr. Reid does with the pending legislation, spokeswoman Beth Levine said.