“I shouldn’t be answering questions of mothers of kindergartners or of children themselves about whether they are safe or hated,” Ms. Dingell said, her voice rising. “We can’t keep going to our corners and not figuring out what we are going to do.”

On the other side of the Capitol, Senator Chuck Schumer, the Democratic leader, has also been pressing Mr. McConnell to take up the House legislation and intends to formally request that the Trump administration cancel its request for $5 billion in border wall funding and reallocate the money for other uses, including initiatives to address gun violence and violent white supremacy, according to a senior aide to Mr. Schumer.

Gun safety has long been one of the most divisive issues in Washington, but this month’s massacres in El Paso and Dayton, Ohio, have thrust it onto the congressional agenda in a way not seen since 2012, after the massacre of 26 children and staff members at the Sandy Hook Elementary School in Newtown, Conn.

Congress failed to take action in the wake of that tragedy; a bipartisan background check bill fell in the Senate in 2013. The House, now under Democratic control, passed a background check bill earlier this year, along with a bill extending the time the F.B.I. has to conduct background checks to 10 business days from three. But Mr. McConnell had refused to take up the measures because President Trump threatened to veto them.

But while Mr. Trump has said repeatedly that Mr. McConnell favors expanding background checks, the leader has not committed to taking up the bill, and he has said he has no intention of bringing the Senate back early to consider it.