The mass shooting at the Washington Navy Yard — which took place a mile from the Capitol — has advocates once again talking about gun legislation, but still without the votes to pass it.

During his weekly news conference, Senate Majority Leader Harry Reid said a background check bill, first drafted in response to the elementary-school shooting in Newtown, Conn., isn’t yet achievable.

“We don’t have the votes,” Reid said. “I hope we get them, but we don’t have them now.”

The Nevada Democrat said he might be willing to move a mental-illness gun bill without a background check expansion, but that comment was quickly walked back by an aide who sent around guidance that Reid does not, in fact, intend to pass a mental-illness gun bill without expanded background checks.

In April, the Senate was five votes short of clearing a 60-vote threshold on a measure from Sens. Joe Manchin III, D-W.Va., and Patrick J. Toomey, R-Pa., to expand background checks.

Its companion measure in the Republican-led House hasn’t gone anywhere.

Rep. Mike Thompson, D-Calif., introduced the House companion background check bill with Republican Rep. Peter T. King of New York earlier this year. Thompson said Tuesday morning that “Congress cannot keep standing on the sidelines while more and more lives are senselessly cut short by gun violence.”

But Rep. Steny H. Hoyer, the No. 2 Democrat in the House, told reporters that while the Navy Yard shooting could bring the gun discussion back, little is likely to be accomplished.

“I’m sure it will renew the debate as to what policy we pursue to not necessarily eliminate — because I’m not sure that’s possible — but to substantially reduce the gun violence,” the Maryland Democrat said.

The House minority whip said the politics remain difficult.

“Yes, I think it will bring up debate. Whether it will bring up action is problematic,” Hoyer said. “If ‘the past is prologue,’ our prologue is not very hopeful.”

Hoyer said many guns rights activists are single-issuer voters, outside groups hold “great sway, particularly in the Republican party” and support for more restrictive gun laws doesn’t “manifest itself at the polls.”

The combative Rep. Steve Stockman, R-Texas, took a different tone on Twitter than the conciliatory approach most politicians used to express sadness for the victims of the Navy Yard shooting.

“12 people tragically killed and self-serving anti-gun extremists trampled each other to beclown and embarrass themselves again. Shameful,” @SteveWorks4You tweeted.

“Better idea: A background check and ‘cooling off period’ before mainstream media can file stories,” another tweet from Stockman’s account read. His press secretary, Donny Ferguson (@DonnyFerguson), tweeted the same thing and controls Stockman’s account.

But Stockman was the exception, not the rule.

Sen. Dianne Feinstein, D-Calif., who has been a strong advocate of an assault weapons ban, said the Navy Yard shooting was one more event to add to the “litany of massacres” that occur when “a deranged person or grievance killer is able to obtain multiple weapons — including a military-style assault rifle — and kill many people in a short amount of time.

“When will enough be enough?” Feinstein asked.

However, the FBI said Tuesday afternoon that it doesn’t have any information that the shooter had an assault rifle in his possession.

Sen. Christopher S. Murphy, D-Conn., who has been helping lead the background check push, vowed to keep up the effort.

“I don’t know what will happen in the coming days, but Americans are not going to sit by and allow these mass shootings while we say absolutely nothing here,” he said. “It’s unconscionable that we sit by and do nothing in Washington as 6,000 people have died across the country since Newtown, 13 more yesterday.”

King, the Republican co-sponsor of Thompson’s gun bill, said in a very brief statement to CQ Roll Call relayed by his spokesman that he was not ready to discuss next steps.

“In light of yesterday’s tragedy, it is too early,” King said.

The White House, meanwhile, said President Barack Obama continues to push the background check bill.

The president “doesn’t accept that it’s the new normal,” White House Press Secretary Jay Carney said of mass shootings. “He believes that Americans don’t and can’t accept that. … We continue to call on Congress to listen to the voices of their constituents and legislate accordingly.”

He said rather than asking the president what he plans to do next, reporters should ask Republicans.

“The problem here is not Democrats,” he said. “The problem here is senators — overwhelmingly from one party — who refuse to do something very simple, which is expand the background check system that everyone believes functions well but needs to function better.”

Meredith Shiner and Niels Lesniewski contributed to this report.