Republican lawmakers, for their part, argue that Mr. Rosenstein’s department has slow-walked important requests and withheld crucial details from documents they do turn over — material they say is necessary to doing their jobs. And their threats are hardly veiled.

“Despite his repeated promises to cooperate, Mr. Rosenstein’s supervision of the Department of Justice has been sorely inadequate,” said Representative Mark Meadows, Republican of North Carolina and one of Mr. Rosenstein’s most outspoken antagonists. “Valid investigative requests from Congress have been slow-walked, stonewalled and impeded at each step of the way under his watch.”

He added, “If Mr. Rosenstein’s hesitance to produce documents and information to Congress represented an effort to save the Department of Justice from embarrassment, it is too late.”

Mr. Rosenstein, aware of the threats against him, has taken unusual steps to try to meet the demands, adding employees to review the requested files and sharing unredacted documents normally off limits to Congress — including memos drafted by the former F.B.I. director James B. Comey about his interactions with Mr. Trump. The department has even set up office space at its headquarters for congressional staff members and lawmakers to review hundreds of thousands of documents already studied by the department’s inspector general, according to a department official.

Those efforts have placated powerful Republican committee chairmen.

After Representative Devin Nunes of California, the chairman of the House Intelligence Committee, threatened last month to hold Mr. Rosenstein in contempt of Congress or proceed with impeachment, Mr. Rosenstein gave him access to an almost completely unredacted F.B.I. memo on the opening of the Russia investigation and won his thanks.

He reached an agreement last week with the two Republicans who run the committees that conduct oversight of the Justice Department, Representatives Robert W. Goodlatte of Virginia and Trey Gowdy of South Carolina, to satisfy the last of their demands for documents related to the investigation into Hillary Clinton’s emails and other decisions related to the Russia case.

But those compromises may have only emboldened Mr. Trump’s fiercest allies, including Mr. Meadows, the chairman of the hard-line House Freedom Caucus, and Representative Jim Jordan of Ohio, a former chairman of the caucus. In an unusual show of defiance, both men have insisted that the agreement with the chairmen of the House Judiciary and House Oversight committees is not good enough and that they need access to an unredacted version of an August 2017 memo outlining the scope of Mr. Mueller’s investigation.