“I have never had any relationship with the Russian government or any Russian officials. I was never in contact with anyone, or directed anyone to be in contact with anyone,” he said.

“On the ‘Russian hacking of the D.N.C.,’” he said, “my only knowledge of it is what I have read in the papers.”

The decision to open the investigations was not based on a dossier of salacious, uncorroborated allegations that were compiled by a former British spy working for a Washington research firm. The F.B.I. is also examining the allegations in that dossier, and a summary of its contents was provided to Mr. Trump earlier this month.

Representatives of the agencies involved declined to comment. Of the half-dozen current and former officials who confirmed the existence of the investigations, some said they were providing information because they feared the new administration would obstruct their efforts. All spoke on condition of anonymity because they were not authorized to discuss the cases.

Numerous news outlets, including The New York Times, have reported on the F.B.I. investigations into Mr. Trump’s advisers. BBC and then McClatchy revealed the existence of a multiagency working group to coordinate investigations across the government.

The continuing investigation again puts the F.B.I. director, James B. Comey, in the middle of a politically fraught investigation. Democrats have sharply criticized Mr. Comey’s handling of the investigation into Hillary Clinton’s use of a private email server. Mrs. Clinton has said his decision to reveal the existence of new emails late in the campaign cost her the election.

The F.B.I. investigation into Mr. Manafort began last spring, and was an outgrowth of a criminal investigation into his work for a pro-Russian political party in Ukraine and for the country’s former president, Viktor F. Yanukovych. In August, The Times reported that Mr. Manafort’s name had surfaced in a secret ledger that showed he had been paid millions in undisclosed cash payments. The Associated Press has reported that his work for Ukraine included a secret lobbying effort in Washington aimed at influencing American news organizations and government officials.