The intelligence agencies already have a degree of unease over the Justice Department’s ability to keep the identity of sources secret. The name of the F.B.I. informant involved in the initial investigation of the Trump campaign’s contacts with Russia was inadvertently made public.

“If you compromise agents, lives can be lost. That is why this is so sensitive,” Senator Angus King, the Maine independent who is a member of the Senate Intelligence Committee, said in an interview. “It is important to be exceedingly careful in this area. That is my only concern, and I hope Mr. Barr realizes that.”

Mr. Barr asked Mr. Trump for the declassification authority, a Justice Department official said on Thursday. He has not detailed the information he is seeking, but some officials have said he is interested in how the C.I.A. concluded Mr. Putin ordered the interference campaign in 2016.

The most prominent of the C.I.A.’s sources of intelligence on Russia’s election interference was a person close to Mr. Putin who provided information about his involvement, former officials have said. The source turned over evidence for one of the last major intelligence conclusions that President Barack Obama made public before leaving office: that Mr. Putin himself was behind the Russia hack.

Long nurtured by the C.I.A., the source rose to a position that enabled the informant to provide key information in 2016 about the Russian leadership’s role in the interference campaign, the officials said.

John O. Brennan, the C.I.A. director under Mr. Obama, would bring reports from the source directly to the White House, keeping them out of the president’s daily intelligence briefing for fear that the briefing document was too widely disseminated, according to the officials. Instead, he would place them in an envelope for Mr. Obama and a tiny circle of aides to read.

But Mr. Trump’s promise to declassify a broad swath of documents suggests that Mr. Barr’s mandate is more extensive than investigating any single source. Mr. Trump’s comments mentioning Britain and Australia appeared to be a reference to the F.B.I.’s investigation of George Papadopoulos, a former Trump campaign aide.