Law enforcement officials from major cities like New York and Los Angeles, where strict gun control laws are aimed at handguns, warned that the bill would usurp states’ authority to set their own laws and effectively impose the lax laws of Southern and rural states on densely populated cities.

Republicans in the Senate would need to pick up at least eight Democrats to pass the measure. And while several Democrats backed a similar measure when it was last voted on in 2013, the politics surrounding guns have shifted since then amid a spate of deadly mass shootings. The result has been a virtual deadlock as Democrats and Republicans cannot agree on how, if at all, to address gun violence.

Several Democrats who voted for the 2013 measure, including Senators Martin Heinrich of New Mexico, Mark Warner of Virginia and Tom Udall of New Mexico, said this week they would not do so this time around. Even Democrats perceived to be the most in favor of gun rights, including Senators Joe Manchin III of West Virginia and Jon Tester of Montana, were cautious about staking out a position before they needed to.

Republican leaders, wary of seeing the measure once again fail on the floor of the Senate floor, were not rushing the concealed-weapon bill to that chamber. Senator John Cornyn of Texas, the No. 2 Republican and a co-sponsor of both the concealed-carry and background check bills in the Senate, said on Tuesday that he was “realistic enough” to realize that following the House’s lead by combining the bills would also be pointless in his chamber.

“If you put them together, it makes it harder to do what we can do and can do now and need to do,” he said.

Senators from both parties view the background check bill as one of the hopeful — albeit narrow — areas of consensus. It was developed in response to a lapse that allowed the gunman in the Sutherland Springs shooting to buy his weapons. The Air Force failed to send his domestic violence conviction to the national database. Had it done so, the gunman, Devin P. Kelley, would have been barred from buying a firearm from a licensed gun dealer. The measure incentivizes states and federal agencies to report criminal offenses and other information to the National Instant Criminal Background Check System.

During the Senate Judiciary hearing, the acting director of the Bureau of Alcohol, Tobacco, Firearms and Explosives also told senators that the agency expected to begin regulating — and could even ban — so-called bump stocks, which can turn semiautomatic rifles into weapons capable of firing long, deadly bursts. The Las Vegas gunman used such devices during his deadly rampage.