No evidence has emerged publicly that Mr. Trump was secretly in contact with or took direction from Russian government officials. An F.B.I. spokeswoman and a spokesman for the special counsel’s office both declined to comment.

Rudolph W. Giuliani, a lawyer for the president, sought to play down the significance of the investigation. “The fact that it goes back a year and a half and nothing came of it that showed a breach of national security means they found nothing,” Mr. Giuliani said on Friday, though he acknowledged that he had no insight into the inquiry.

The cloud of the Russia investigation has hung over Mr. Trump since even before he took office, though he has long vigorously denied any illicit connection to Moscow. The obstruction inquiry, revealed by The Washington Post a few weeks after Mr. Mueller was appointed, represented a direct threat that he was unable to simply brush off as an overzealous examination of a handful of advisers. But few details have been made public about the counterintelligence aspect of the investigation.

The decision to investigate Mr. Trump himself was an aggressive move by F.B.I. officials who were confronting the chaotic aftermath of the firing of Mr. Comey and enduring the president’s verbal assaults on the Russia investigation as a “witch hunt.”

A vigorous debate has taken shape among some former law enforcement officials outside the case over whether F.B.I. investigators overreacted in opening the counterintelligence inquiry during a tumultuous period at the Justice Department. Other former officials noted that those critics were not privy to all of the evidence and argued that sitting on it would have been an abdication of duty.

The F.B.I. conducts two types of inquiries, criminal and counterintelligence investigations. Unlike criminal investigations, which are typically aimed at solving a crime and can result in arrests and convictions, counterintelligence inquiries are generally fact-finding missions to understand what a foreign power is doing and to stop any anti-American activity, like thefts of United States government secrets or covert efforts to influence policy. In most cases, the investigations are carried out quietly, sometimes for years. Often, they result in no arrests.