So Mr. Mueller, a former F.B.I. director and Marine Corps officer known for his self-discipline and meticulousness, has chosen to let his work speak for him. In 16 months, he has produced indictments against three companies and 34 people, including a group of 12 suspected Russian intelligence agents; last month, his work led to a conviction, on eight counts, of President Trump’s former campaign manager, Paul Manafort.

In June, Mr. Mueller’s approval rating hit a low: A Politico poll showed that 36 percent of registered voters viewed him unfavorably, up from 23 percent a year earlier. His reputation has improved since then: A poll released this week by Quinnipiac University found that 55 percent of voters said he was conducting a fair investigation; 32 percent said he was not.

Still, the offensive against the special counsel is unlikely to stop.

In May last year, the day after Mr. Mueller assumed his role, Mr. Trump wrote on Twitter: “This is the single greatest witch hunt of a politician in American history!” The president’s use of “witch hunt” has lately skyrocketed, and he recently exhorted his online followers to “study the late Joseph McCarthy, because we are now in period with Mueller and his gang that make Joseph McCarthy look like a baby!”

Sean Hannity of Fox News made the “Mueller Crime Family” a mantra for his nearly four million nightly viewers, the biggest audience in cable news. The conspiracy theorist Alex Jones of Infowars also joined in, suggesting with no evidence that the special counsel covered up a pedophilia ring and pantomimed shooting Mr. Mueller with a pistol.

Since joining the president’s legal team in April, Mr. Giuliani has been an especially sharp critic, telling Mr. Hannity: “You know how sometimes the cover-up is worse than the crime? In this case, the investigation was much worse than the no-crime.” He has been explicit about the purpose of these attacks: to discredit the investigation in the eyes of the public and make it less likely that Congress will act on its findings.

“What we’re doing here, it’s public opinion,” Mr. Giuliani told CNN in May. “Because eventually the decision here is going to be impeach or not impeach.”