This week, the panel passed a measure that would expand the background checks to private gun sales, and another measure to renew a grant program to help schools improve security.

The background check bill is expected to be substituted or amended by its sponsor, Senator Charles E. Schumer, Democrat of New York, to attract the support of more Republicans. In theory, it is a measure that both parties can support, but it is ensnared over a debate over record-keeping that may undermine it on the floor.

The committee vote on Thursday to approve the assault weapons ban was 10 to 8, along party lines.

Senator John Cornyn, Republican of Texas, offered amendments to the bill that would have created exemptions from the ban for female victims of violent crimes, those who had received a protection order, and residents near the Southwest border and in rural areas. Those amendments all failed.

In debating the bills, the committee laid bare the essence and emotions of the debate over how to prevent gun violence and the meaning of the Second Amendment, a fight that is likely to continue on the Senate floor.

The most testy exchange occurred between Ms. Feinstein and Senator Ted Cruz, Republican of Texas, who challenged Ms. Feinstein on her constitutional knowledge, asking her if she would apply regulations to the First and Fourth Amendments similar to those she was seeking on firearms ownership.

Ms. Feinstein, whose career has been shaded with episodes of gun violence, would not brook it. “I’m not a sixth grader,” she said. “I’m not a lawyer, but after 20 years I’ve been up close and personal with the Constitution. I have great respect for it.”

If the Senate passes even modest measures next month, they will face a steep climb in the House. “I’ve made it perfectly clear if the Senate passes a bill, we will be happy to review it,” Speaker John A. Boehner, Republican of Ohio, said in an interview on Thursday. “In the meantime, our committees are continuing to have hearings on this issue, continuing to look at our violent society and the causes of it and what we can do to reduce the incidents of violence in our society.”

President Obama has made an emphatic call for new gun regulations, but he so far has not spent extensive political capital on the effort. In visits to Capitol Hill to meet with lawmakers this week, the issue barely came up. “The Senate has now advanced legislation addressing three of the most important elements of my proposal to help reduce the epidemic of gun violence in this country,” he said in a statement.