In the campaign’s early months in 2015, Mr. Trump publicly predicted that he would “get along very well” with Vladimir V. Putin, the Russian president. And at a National Rifle Association event that April, Mr. Trump met Alexander Torshin, the deputy governor of the Russian central bank who has been linked to both Russia’s security services and organized crime, according to Mr. Torshin.

Mr. Torshin would later try to broker a “backdoor” meeting with the campaign. Mr. Mueller and congressional investigators have obtained emails relating to the campaign’s contacts with Mr. Torshin.

Mr. Nunberg has been asked to provide information from Nov. 1, 2015, through the present about Mr. Trump and several of his aides, including his communications director, Hope Hicks; Corey Lewandowski, his former campaign manager; Stephen K. Bannon, the president’s former chief strategist; Mr. Trump’s longtime bodyguard, Keith Schiller; and Michael D. Cohen, the former Trump Organization lawyer.

In a somewhat Trumpian fashion, Mr. Nunberg, in an interview, seemed more interested in the spectacle he created this week than the legal ramifications of vowing to tear up a subpoena from Mr. Mueller’s team on live television.

“I do believe that Donald Trump and Roger Stone are right: No publicity is bad publicity,” Mr. Nunberg said. “To me, it made for good TV, and everybody’s talking about me.”

(He added that Gary D. Cohn’s departure from the White House had probably bumped him from the news.)

Mr. Nunberg described his TV appearances on Monday as “just me letting loose.” He was red-faced, defiant and nonchalant about being asked to provide documents and appear before the special counsel. Erin Burnett, the CNN anchor, asked him if he had been drinking, because she smelled alcohol on his breath. He denied it.

Several people close to Mr. Nunberg said that he had struggled with sobriety on and off, and that contact from Mr. Mueller’s team had distressed him. According to two friends of Mr. Nunberg, who spoke on the condition of anonymity because they did not want to discuss his personal life, Mr. Nunberg had appeared out of sorts in TV appearances for weeks. In January, he laughed off a bizarre and combative appearance on MSNBC by telling a friend who had asked him if he was drunk that he had been on cold medicine.