Many Democrats, and some Senate Republicans, believe the only legislation that has a whisper of a chance of passing would be bills that are tightly focused on more consensus elements like enhancing background checks or limits on magazines, subjected to debate in committee and then brought to a vote after building bipartisan support.

That would be a departure from recent years, when the leadership often sidestepped committees and sought to take fights directly to the floor.

Others, particularly those senators who have long fought for gun control measures, believe a plodding process allows too much time for opposition to build, and prefer to fast-track measures by adding them as amendments to other bills, even blocking bills in ways that have angered Democrats, until they are granted votes on those ideas.

“We can’t sit around for months talking and letting the gun lobby run out the clock,” said Senator Frank R. Lautenberg, Democrat of New Jersey. “If we’re going to make progress, it’s essential that we move quickly and start voting as soon as possible.”