Mr. Brady’s demands soon prompted a tense confrontation with F.B.I. officials at the bureau’s headquarters in Washington. The meeting was mediated by Seth D. DuCharme, now the acting U.S. attorney in Brooklyn and at the time a trusted aide and ally of Mr. Barr’s at the Justice Department in Washington.

The F.B.I. viewed the investigative steps into Mr. Biden that Mr. Brady sought as unwarranted because the Delaware inquiry involving money laundering had fizzled out and because they were skeptical of Mr. Giuliani’s material. For example, they had already examined a laptop owned by Mr. Biden and an external hard drive that had been abandoned at a computer store in Wilmington and found nothing to advance the inquiry.

Investigators were also worried that the effort might become public — particularly any interviews of witnesses who were Mr. Giuliani’s sources of information — and could drag the F.B.I. back into presidential campaign politics, the same turbulent path it had stumbled down in 2016 with investigations into Hillary Clinton’s private email server and the Trump campaign’s ties to Russia.