“Ultimately, I think we’re going to have to call the Republicans’ bluff,” Senator Chris Murphy, Democrat of Connecticut, said, if a breakthrough cannot be reached soon. Democrats in the Senate believe that Republicans could not withstand the pressure of a straight vote on expanded background checks, which have the support of as much as 90 percent of the public.

On immigration, a bipartisan group of eight senators hopes to unveil legislation that would strengthen border security, establish a new guest-worker program for foreign workers and establish a pathway to citizenship for the estimated 11 million illegal immigrants already in the country. That, too, could slip to next week amid hard negotiations, especially over the scope of the guest-worker program.

Senator Patrick J. Leahy, the Vermont Democrat who is chairman of the Judiciary Committee, said the window for moving on immigration was narrowing, and if the group of eight senators could not reach agreement in the next week or two, he would have to begin a more formal legislative process in his committee.

“This is one where the senators are going to have to stand up and vote yes or vote no,” he said.

Adding to the tension and emotions of the issues, the families of children and educators killed at Sandy Hook in Connecticut will converge in Washington for private meetings with senators on Tuesday, Wednesday and Thursday. Rallies are also in the works for immigrant populations pushing for change. Such shows of force helped galvanize legislative drives on the immigration issue in 2006 and 2007, but they were not enough to get hard-fought legislation through the Senate.

On the fiscal front, President Obama’s budget release on Wednesday will add a third set of tax-and-spending plans to a Republican version that has passed the House and a Democratic one that passed the Senate. That should kick off talks to try to find some common ground and reach a deficit deal that would encompass changes to entitlement programs like Social Security and Medicare and an effort to overhaul and simplify the tax code.