In a sprawling speech to Congress last month laying out ­dozens of proposals, President Biden set one specific deadline for ­law­makers: to deliver a long-discussed overhaul of police practices meant to stem the killings of Black citizens at the hands of law enforcement officers.

“Let’s get it done next month, by the first anniversary of George Floyd’s death,” he said on April 28, one week after a jury convicted former Minneapolis police officer Derek Chauvin of murdering Floyd. “The country supports this reform and Congress should act.”

That deadline — May 25 — is now set to come and go without action. Although talks continue productively, according to participants, no deal is in immediate sight and the House is leaving Washington on Friday for a three-week break.

Key lawmakers this week played down the passing milestone — “I’m not really looking at a schedule. I’m just trying to get this bill done right,” Sen. Cory Booker (D-N.J.), a lead negotiator, said Wednesday — but the lapse has left them in a precarious position.

On Capitol Hill, successful legislating tends to be deadline-driven — pushed along by the expiration of existing laws or mounting political pressure. And without a clear timeline, momentum could drain away from the talks, perhaps extending them into the midterm election season, when lawmakers tend to become less amenable to compromise.

According to several lawmakers and aides familiar with the state of the talks, progress has been slow on the most sensitive matter: whether to loosen or eliminate the doctrine of “qualified immunity” that shields police officers and departments from civil liability in cases of misconduct.

Democrats have made eliminating — or at least loosening — the doctrine a cornerstone of their overhaul efforts. Republicans by and large have resisted making any changes, fearful that exposing police officers to lawsuits could cause them to adopt less aggressive and less effective tactics.

Sen. Tim Scott (S.C.), the lead Republican negotiator, has worked with Democrats on a possible compromise that would maintain immunity for individual police officers but allow those alleging gross misconduct to sue police departments instead. But those familiar with the talks said it was not yet clear that there was a path to a deal.

Members of both parties said this week that they expect other matters under discussion — including the banning of choke holds, the creation of a national database of suspect officers and the imposition of new training standards — to quickly fall into place if there is an accord on qualified immunity.

But multiple Republican senators this week expressed deep skepticism about the prospect of rolling back the doctrine and exposing officers or the jurisdictions that hire them to new litigation. Many lawmakers met with police officials last week — the annual National Police Week, which typically attracts thousands of officers to Washington for a yearly memorial service.

“I can tell you, a lot of law enforcement agencies are very concerned about it and what it would mean for their bottom line,” said Sen. Marco Rubio ­(R-Fla.). “What they’re saying is we’re now going to get sued every time we pull somebody over. . . . We’re just going to have hundreds and hundreds of lawsuits, and we’re going to hire an army of lawyers.”

Sen. John Neely Kennedy (R-La.) said his “strong bias” was to leave the doctrine in place, while Sen. John Cornyn (R-Tex.) said he believed that the Chauvin conviction showed that avenues were already in place to deter bad cops.

“I think there needs to be accountability, but as we’ve seen in the George Floyd killing, there is accountability, and it’s called criminal prosecution,” Cornyn said. “I don’t want to make it harder on line law enforcement officers to do their job worrying about their pension or their ability to provide for their family, and so I’m very reluctant to open up a new avenue for suing for money damages.”

Senate Minority Leader Mitch McConnell (R-Ky.) said earlier this month during an appearance in Scottsville, Ky., that continuing qualified immunity is “essential.”

“I don’t know how we get anybody to be a police officer” without it, McConnell said.

But advocates for eliminating qualified immunity say allowing it to remain will leave victims of misconduct without a key method to seek accountability.

“Qualified immunity isn’t saying police officers won’t get their day in court . . . but it will let Black families have their day in court, too,” said Ben Crump, a lawyer for the families of several victims of police violence, after leaving a recent meeting with Senate Majority Leader Charles E. Schumer (D-N.Y.). “All we want is everybody to be able to have equal justice under the law.”

There are some reasons for optimism. Several Republican senators said they would reserve judgment while the negotiations continued, and many of them — including McConnell — have expressed confidence in Scott to negotiate a workable deal.

Scott on Tuesday acknowledged that changes to qualified immunity remained a “work in progress” and said he was open to changes in only the most egregious cases of misconduct.

“The bottom line is a very simple one: In cases of serious bodily injury or death, there could be more exposure if your officers have done something that is in violation that has led to that death,” he said. “Certainly they’d be more exposed.”

The national Fraternal Order of Police, meanwhile, has not rejected the Scott compromise outright, while remaining squarely against allowing lawsuits against individual officers.

“We’re a union, and it’s our responsibility to ensure that to the fullest extent possible, that officers are indemnified for actions they take under the color of law,” said Jim Pasco, the executive director of the national FOP. “I leave it to departments and municipalities to decide whether they want to shoulder the burden of any potential lawsuits.”

Leading Democrats involved in the talks, including Booker and Rep. Karen Bass (D-Calif.), have not publicly responded to Scott’s proposal and have sounded an upbeat note on the talks. They are also dealing with other pressures — from advocates and activist groups pushing for thorough changes not only to qualified immunity but also to federal civil rights laws to make it easier for the Justice Department to pursue prosecutions of officers.

“What’s important is that we’re on track to get it done right,” Bass said Wednesday. “The biggest holdup is that we just haven’t had enough time.”

Other pitfalls abound, including a backdrop of rising crime — particularly a spike in big-city homicides — that has driven rising anxiety about the wisdom of putting new fetters on police powers.

“What people don’t rely on enough around here is common sense, and the bottom line is, yes, if criminals know the police department can’t chase them, can’t be aggressive, we’re likely to see violent criminals become more emboldened,” Rubio said.

Sen. Josh Hawley (R-Mo.) said Wednesday that he would be inclined to support a policing overhaul only if it were accompanied by an infusion of federal funding for police — a proposal that could put Democrats in a difficult position if Republicans decided to rally behind it. Although Democrats a generation ago supported federal spending to put more police on the street, now many on the party’s left wing have advocated for diverting funds away from police departments and toward social service and mental health agencies that they say are better equipped to deal with the root causes of crime.

“We ought to do a lot on police, but I think we ought to increase their funding and put more cops on the street,” Hawley said. “I imagine [Democrats] wouldn’t support that. But I think that’s the need of the hour.”

The advocates involved in the talks, meanwhile, are keeping their powder dry and hoping for the best. NAACP President Derrick Johnson called for quick action after the Chauvin verdict, but a spokesman said he continues to support the talks.

“While we hoped for the police reform bill to be signed into law by May 25, President Johnson wants to see the right bill, not a rushed bill,” said Jonah Bryson, a NAACP spokesman.

And the FOP, whose support will be crucial to winning broader support among lawmakers, remains engaged. Pasco, who said he has been in daily contact with the principal negotiators, said he has been encouraged by the gradual progress toward “legislation that will make this country safer and more inclusive for all citizens and for police officers.”

“Where we’re going to end up, I don’t know,” he said. “But I will say this: As long as we’re still talking, there’s still hope.”

Seung Min Kim contributed to this report.