WASHINGTON — Susan Rice, the White House domestic policy adviser, called the leaders of the nation’s largest policing groups last month to promise a significant reset in their relationship as the Biden administration finished an executive order on police reform, a move that averted a potential breach that had been brewing for months, according to several people briefed on the calls.

The groups welcomed Ms. Rice’s outreach, which amounted to a vow to incorporate more of their thinking in the order and possibly an implicit mea culpa. The White House had solicited input from the groups, but had not engaged with them on substance and details; their frustrations only soared in the days before her phone calls, when they were blindsided by the leak of an 18-page draft executive order that contained language they found objectionable.

Ms. Rice’s course correction dovetails with a broader shift in the White House toward a more centrist stance on policing as violent crime rises. And it underscores President Biden’s struggle to satisfy civil-rights groups, whose cries for reform hit a fever pitch after the murder of George Floyd at the hands of the police in May 2020, while blunting critics who say Democrats are soft on crime.

This more centrist view is likely to be on display when Mr. Biden meets on Thursday with the newly elected mayor of New York City, Eric Adams, a former police captain whose ascent illustrates how far Democrats have moved to the center on criminal justice issues, just two years after progressives began calling to defund the police.

In an interview, Ms. Rice sought to bridge that gap, saying that Mr. Biden’s recognition that responsible, respectful policing is essential for effective public safety has not undermined his longtime support for law enforcement.

“Yes, we need police on the streets, well equipped, but we need them to have the cooperation and trust of the community. These things are not in opposition — they are mutually reinforcing,” Ms. Rice said.

She also noted that the draft executive order was not close to being done, and that many issues remained unresolved.

One sticking point is a guideline that could push federal officers, and most likely the state and local police, to tighten their use-of-force standard, which regulates when officers can shoot. Civil liberties groups have lauded the change, but police leaders have said they cannot abide it.

Chuck Wexler, the executive director of the Police Executive Research Forum, which advises departments on best practices, said that he had seen “significant breakthroughs” in communications with the White House in recent weeks.

“We are not in opposition to reforms,” he said. “We wanted to make sure that the executive order balanced the need for police reform with the changing nature of crime and policing happening across the country.”

The roots of the order began with the murder of George Floyd by a Minneapolis police officer and the subsequent calls for reforms to address issues such as racism in policing and the use of lethal force.

As it became clear in September that a bill known as the George Floyd Justice in Policing Act would fail to pass the Senate, the White House began looking more closely at addressing some of the issues via an executive order. Ms. Rice and her team led its development.

In late summer, according to people familiar with the process, the White House began holding conference calls with groups it had spoken with about the Floyd legislation, including leaders of police groups — many of whom had endorsed aspects, but not all, of the bill — as well as civil rights groups and representatives of families of people killed by the police.

Those early calls included two listening sessions with law enforcement groups in late October: one with Terrence M. Cunningham, the deputy executive director of the International Association of Chiefs of Police, and Jim Pasco, the executive director of the Fraternal Order of Police and a longtime friend of Mr. Biden; and another with a broader group of similar organizations, an official said.

The White House held about 20 meetings with various law enforcement groups from August to December, according to one official. But police leaders told members of Congress and senior law enforcement officials that the engagement seemed perfunctory.

Ms. Rice challenged that view, describing the meetings as part of a planned listening phase and said that officials intended to engage more deeply over draft language later.

Multiple advocates said the White House’s meetings with civil-rights groups last year had also been “listening” style meetings.

As the draft process unfolded, officials said, the White House was separately warned that it needed to engage more with police leaders in order to secure their support for the final order. Among those making that case were two Democratic senators — Richard J. Durbin of Illinois and Cory Booker of New Jersey, who had both worked on the George Floyd bill — as well as Alejandro N. Mayorkas, the secretary of homeland security, and the No. 2 and No. 3 officials at the Justice Department, Lisa O. Monaco and Vanita Gupta.

During an early November meeting, for example, Mr. Durbin and Mr. Booker said that their districts were grappling with rising crime, and that it would be a practical and political problem to be at odds with the police, according to people briefed on that meeting.

But the White House did not shift its approach, and in late December it circulated a draft of its executive order to other executive branch agencies for comment. Blurry images of that draft leaked, and a copy was published on Jan. 5 by The Federalist, a conservative website.

Enraged law enforcement groups especially disliked the tenor of the order’s policy preamble, which spoke of “systemic racism” in the criminal justice system.

Mr. Pasco said he told the Justice Department and the White House that issuing that version of the order “would cause an irreparable rift between Biden and the police.” Another group leader told the administration the headline would be “Biden turns his back on police.”

That raised alarms. Mr. Biden has been mindful of his relationship with the police, especially since major police unions with which he had previously worked with endorsed President Donald J. Trump during the 2020 election.

Some officials have said they understood that the draft was nearly ready for publication when it leaked. But White House officials have countered that impression. Dana A. Remus, the White House counsel, called it a “very early draft” that was not close to being ready.

Either way, its publication led Ms. Rice to make conciliatory phone calls with an eye toward more substantive discussions.

Since then, Ms. Rice said that law enforcement, civil rights groups and others had shared their reactions and officials were “trying to be responsive to what we’ve heard.”

Mr. Pasco said the leaked order featured provisions everyone could agree on, like standardizing and improving credentialing for police agencies; creating a national registry of police officers who have been fired for cause — after due-process hearings — so those officers are not rehired by other departments; tightening restrictions on when the police can use so-called no-knock warrants in raids; and banning the transfer to the police of military equipment like flash-bang grenades.

But a section about using force remains a point of contention. Under current law, officers may shoot if they fear for their lives or those of people around them. The draft order allows for deadly force only “as a last resort when there is no reasonable alternative.”

The policy change and others in the order apply only to federal law enforcement, but the state and local police could be encouraged to also adopt the changes because of a provision about federal discretionary grants. (Discretionary grants make up a small portion of the billions of dollars that Congress earmarks for local law enforcement.)

Earlier versions of the order had explicitly called for making such grants conditional on adopting the new policies, according to officials who worked on the draft. But the leaked version is softer, saying that the money should be distributed “in a manner that furthers the policy goals,” like the use-of-force standard.

Civil-rights groups are insisting that the use-of-force language remain in the final order. Udi Ofer, the deputy national political director of the American Civil Liberties Union, said that Mr. Biden had “the authority to bring to life a strong standard that will save lives.”

But Mr. Pasco portrayed the provision as a deal breaker, arguing that it would open the door for “hindsighted second-guessing” of officers. He said that the White House should instead focus on ideas for which there was consensus.

It remains unclear what the administration will do. For now, however, police group leaders say they have the opportunity to make their case more strongly.

“Crime is an issue and I think the president recognizes that,” Mr. Wexler said. “The pendulum is swinging back towards crime being a significant priority.”