The CNN report relied on two unnamed sources who described an email purporting to show that Mr. Trump and other campaign officials had received advance notice about a cache of hacked Democratic documents that were about to be posted by WikiLeaks, the anti-secrecy group. In fact, the email had been sent the day after the cache was posted publicly. CNN later corrected its story.

“These disturbing circumstances warrant examination,” Mr. Futerfas concluded in his letter.

A spokesman for Mr. Schiff, Patrick Boland, denied that the congressman or his staff had leaked classified or confidential information. But, he said, Mr. Schiff reserved the right to speak out about the “noncooperation” of a witness.

“We do not permit witnesses to represent publicly that they are fully cooperating with our committee and privately refuse to answer questions pertinent to our investigation on the basis of meritless claims of privilege,” Mr. Boland said. He added that Mr. Schiff would continue to do so when appropriate.

The New York Times and numerous other outlets reported on Mr. Schiff’s remarks, which were made public after the all-day session concluded. Mr. Schiff said that Mr. Trump had cited attorney-client privilege in declining to provide the committee details of a July telephone conversation with his father about a 2016 meeting at which Trump campaign officials had expected to receive damaging information from the Russian government about Hillary Clinton.

The Times also reported, based on unnamed sources familiar with the testimony, that Mr. Trump had said he could not remember a phone call with a blocked number that records indicate took place as he was setting up the 2016 meeting.

Mr. Trump is set to be back on Capitol Hill on Wednesday for another closed-door session, this time with the Senate Intelligence Committee, according to a person familiar with the schedule.