The proposal follows a series of aggressive changes in posture at the agency since last fall, when Republicans gained a majority on the five-member board.

In early December, a mere two weeks into his tenure, Mr. Robb released a memo announcing the end of many of his predecessors’ initiatives, including a campaign against employers who improperly classify workers as contractors, and featuring a long a list of hot-button issues on which regional directors were required to seek input from his office.

“New general counsels will at some point signal cases they want to look at,” said Wilma B. Liebman, a former chairwoman of the labor board. “But this was so sweeping and so fast that it was just kind of startling.”

That same month, the agency overturned a key Obama-era ruling that had made it easier to hold companies responsible for labor-law violations at companies they do business with, such as franchisees and contractors.

Mr. Robb came to his position after a career largely spent representing management, including handling part of the Reagan administration’s litigation against the air traffic controllers’ union that waged an illegal strike in 1981. Many labor historians say the government’s hard line in firing the controllers contributed to organized labor’s decline in subsequent decades.

The labor board’s general counsel is confirmed by the Senate. The counsel has independent authority as a prosecutor, derived from the National Labor Relations Act, and performs other duties on behalf of the agency’s board, which acts as its highest court of appeals.

Demoting the regional directors — there are 26, including two vacancies — and inserting a new group above them would most likely require board approval. The regional directors’ account suggested that the new officials would probably be civil servants as well, rather than political appointees.

Michael J. Lotito, a lawyer with the management-side firm Littler Mendelson, who has discussed the proposal with officials at the agency, said they had assured him that it was largely a response to budget cuts reflecting a significant decline over several decades in the number of labor charges filed.

He said some of the savings could come from staff reductions among managers and supervisors at the regional offices, achieved in part through attrition.

The agency itself said that “given budgetary issues, the general counsel is assessing the current organizational structure for possible changes,” but added, “No specific plan involving the restructuring of our organization has been developed.”

Labor advocates and even management-side lawyers often praise the professionalism of the regional directors, but critics consider them too sympathetic to workers and unions.

“Some are way more ideologically pro-union than others, but they all tend to be fairly ideological,” Mr. Lotito said. “The agency tends to promote — I think that it should — individuals who want to protect the rights of employees. But if you’ve been doing that all your life, you can miss the rights of the employer.”

The proposed changes appear consistent with a broader Trump administration suspicion of longtime civil servants. President Trump’s former chief strategist, Steve Bannon, called for the “deconstruction of the administrative state.”

During the administration’s first year, dozens of senior career officials resigned or retired from the State Department and the Environmental Protection Agency as many complained of being sidelined or ignored by political overseers.

The Interior Department reassigned a few dozen senior civil servants to posts that often made little use of their expertise. In one case, a top climate policy official was reassigned to the office that collects royalty payments from oil and gas companies. He quit not long after.

According to the N.L.R.B. regional directors’ letter, Mr. Robb said on the conference call on Jan. 11 that the changes were necessary and independent of budget considerations — implying a lack of confidence in the directors’ ability to investigate and adjudicate allegations of labor-rights violations.

Mr. Robb, according to the letter, said the agency might hire a handful of district directors, each with authority over a portion of its 26 regions, and proposed lowering the regional directors’ rank within the Civil Service.

The letter expressed concerns that Mr. Robb was intent on “removing many of the core responsibilities of the sitting regional directors,” though it acknowledged that it was unclear how much authority he intended to shift to district directors, how many there would be, and whether any regional offices would be closed or consolidated.

Mr. Lotito, the management lawyer who has discussed the concept with agency officials, said the idea was to keep most or all of the regional offices in place and allow management and labor to appeal decisions to the district directors.

“The regional directors are their own fiefdoms,” he said. “If there was an ability to go to a district director, who oversees eight regions, chances are that would drive consistency.”

Like the civil servants who were reassigned at the Interior Department, the regional directors at the labor board are members of the Senior Executive Service, which was established by Congress in the late 1970s as an elite corps of executives who could be deployed to tackle the government’s thorniest challenges.

Conservatives and some nonpartisan experts have complained that members of the Senior Executive Service are far less mobile than Congress envisioned, burrowing into particular agencies.

By proposing to downgrade the regional director positions into the ranks of standard government workers, where they would make less money and have narrower authority, Mr. Robb in some respects proposed going further than Interior Secretary Ryan Zinke, who reassigned Senior Executive Service members but preserved their status.

The changes proposed by Mr. Robb “would have a severe and negative impact on our agency and our stakeholders,” the directors wrote.

William Valdez, president of the association representing the senior executives, said the labor board could eliminate such positions, making the regional directors eligible for comparable jobs elsewhere in government, but could not demote them en masse. As a practical matter, he said, many of the regional directors may choose to reapply for their jobs at lower rank.

Even many management-side lawyers consider the regional directors effective. “Some people are better than other others, but I think the standards are pretty high,” said Steven M. Swirsky, a former field lawyer for the labor board who has spent decades representing employers.

At a Jan. 19 meeting with an American Bar Association committee whose members represent both management and labor, Mr. Robb argued that the reaction to his proposal was overblown.

According to someone familiar with the discussion, Mr. Robb acknowledged that he would need the approval of the agency’s board for any changes, and he said he would seek public comment as well as input from the regional directors.

In the meantime, the letter from the regional directors indicates that many may take Mr. Robb’s proposal as a cue to resign or retire, as some of their counterparts at the State Department, the Environmental Protection Agency and the Interior Department did.

“We believe the changes you suggest, including the removal of directors from the Senior Executive Service, will cause senior directors and managers, whose institutional knowledge is a valuable asset to the agency, to retire sooner than they otherwise intended,” the directors wrote to Mr. Robb. “As you can imagine, the information you provided to the regional directors has created much uncertainty and has disheartened us.”