Officials at the department believe that the conservatives have now gone too far with document requests related to continuing investigations that the lawmakers clearly do not support, including the inquiry led by the special counsel, Robert S. Mueller III, into Russia’s election interference. A former federal law enforcement official familiar with the department’s views said that Mr. Rosenstein and top F.B.I. officials have come to suspect that some lawmakers were using their oversight authority to gain intelligence about that investigation so that it could be shared with the White House.
Mr. Trump’s threat on Wednesday to intervene bolstered those voices and could undermine the Justice Department’s ability to protect some of its most closely held secrets. Lawmakers conducting oversight are usually given summaries of the information, but not the intelligence collected directly from wiretaps and sensitive sources.
Similar standoffs between law enforcement officials and Congress have resulted in compromise dating back decades, but in those cases, the Justice Department had the support of the president. Without Mr. Trump’s support, Congress is gaining the advantage.
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.
Democrats fear that the Republican requests — many of which call on the department to ignore longstanding policy about what it shares with Congress — are meant as a trap. Either Mr. Rosenstein can turn over information that could be used to undermine the special counsel’s inquiry, or he could refuse, giving Mr. Trump cover, or even cause, to fire the deputy attorney general.
Representative Jerrold Nadler of New York, the top Democrat on the Judiciary Committee, said the latest Republican efforts were “clearly trying to sabotage” Mr. Mueller’s investigation and court a confrontation with Mr. Rosenstein.
“All of this noise is aimed at undermining the special counsel’s work as the investigation closes in on the president,” Mr. Nadler said in a statement. “The president’s attacks on the Department of Justice grow more paranoid by the day. The case for obstruction of justice — and the complicity of these House Republicans — grows day by day as well.”
Mr. Rosenstein, who has already given the Republican lawmakers access to hundreds of thousands of pages of documents, has made clear in recent days that he does not intend to go further.
On Monday, the Justice Department wrote to Mr. Meadows and Mr. Jordan to deny them access to the document about the scope of the Russia inquiry, citing department policy against sharing information on a continuing investigation.
“The department recognizes the keen interest that Congress has in the special counsel’s investigation, but, respectfully, we must adhere to the longstanding position of the department that congressional inquiries pertaining to ongoing criminal investigations threaten the integrity of those investigations,” Stephen E. Boyd, an assistant attorney general, wrote in the letter, a copy of which was provided to The New York Times.
“We hope you can respect our position,” he added.
And on Tuesday, Mr. Rosenstein, reacting to reports that Mr. Meadows had drafted articles of impeachment to use against him if needed, pushed back hard.
“If we were to just open our doors to allow Congress to come and rummage through the files, that would be a serious infringement on the separation of powers,” Mr. Rosenstein said at an event in Washington. “It might resolve a dispute today, but it would have negative repercussions in the long run, and we have a responsibility to defend the institution.”
It is unusual for rank-and-file members of a committee to challenge or maneuver around their own chairmen on sensitive matters. But Mr. Jordan and Mr. Meadows are known as two of the most confrontational Republicans in the chamber. Mr. Meadows has developed a close relationship with Mr. Trump, while conservatives are talking up Mr. Jordan as a candidate to succeed Representative Paul D. Ryan as speaker.
Neither lawmaker responded on Wednesday to requests for comment.
In an apparent break with both men, Mr. Gowdy, chairman of the Oversight Committee, said on Wednesday that he was “satisfied” that his committee and the Judiciary Committee now had the access it needed to documents relevant to a continuing joint investigation into decisions made by the department in 2016 and 2017.
“I appreciate Rosenstein’s willingness to work with the committees, and I have confidence in his leadership,” Mr. Gowdy said, adding that Mr. Mueller should be given “the time, the independence and the resources to conduct a thorough investigation.”
Mr. Trump, who has privately fumed about Mr. Rosenstein, has said publicly that the deputy attorney general faced conflicts of interest. The president criticized Mr. Rosenstein for signing a search warrant application to permit federal agents to eavesdrop on one of Mr. Trump’s former campaign aides.
Mr. Rosenstein assumed oversight of the investigation and appointed Mr. Mueller as special counsel after the attorney general, Jeff Sessions, recused himself last year. Mr. Trump has repeatedly attacked Mr. Sessions for his recusal.
The president’s warning on Wednesday was a sharp departure from earlier comments, when he has said that although he is frustrated with the investigation into his campaign, he is not supposed to be involved with department matters.
“I am not supposed to be involved with the F.B.I.,” Mr. Trump said in an interview in November. “I’m not supposed to be doing the kind of things that I would love to be doing. And I’m very frustrated by it.”
But Mr. Trump is increasingly on the defensive, after an F.B.I. raid of the office and hotel room of his personal lawyer, Michael D. Cohen, and the disclosure of more than 40 questions that the special counsel would like him to answer. The questions touch on a variety of topics, including coordination with the Russians during the presidential campaign and actions that Mr. Trump has taken as president and whether they were intended to derail the inquiry.