Mr. Trump’s team has been conducting opposition research on Mr. Mueller and his growing staff, searching for points of vulnerability. Already the president and his allies have publicly pointed to some of the lawyers Mr. Mueller has hired on his staff who contributed money to Mrs. Clinton or other Democrats in the past; one represented the Clinton family foundation. Contributions to Democrats have not been a disqualifier for Mr. Trump in putting together his own staff — including his new communications director, Anthony Scaramucci, who donated money to Mrs. Clinton and Barack Obama in the 2008 campaign cycle — but the president and his team now argue that Mr. Mueller’s team is compromised as a result.

Mr. Mueller has not responded to the attacks, and the Justice Department has said any potential conflicts will be dealt with under normal procedures. In using the term “conflict of interest,” Mr. Trump may be laying a predicate for eventually ordering the Justice Department to fire Mr. Mueller. Under current procedures, the special counsel can be dismissed only under certain circumstances, including for conflicts of interest.

Whether Mr. Trump can do to Mr. Mueller what the Clintons did to Mr. Starr remains an open question. The Clintons did not go on the attack when the first special counsel, Robert Fiske, was picked to look into their land dealings. They shifted strategy only after Mr. Starr was named to replace Mr. Fiske by a three-judge panel led by a conservative judge who, shortly before the appointment, had lunch with two Republican senators who were harsh critics of the Clintons.

Though Mr. Starr, a former solicitor general and appeals court judge, was widely respected before becoming independent counsel, he was seen as a conservative whose partiality was immediately questioned. Over the course of his investigation, he made judgment choices that gave ammunition to the Clintons as they assailed his handling of the case.

Mr. Mueller, by contrast, is not seen as a political figure. A Marine veteran decorated for combat in the Vietnam War, he was a career prosecutor who worked his way up to head the F.B.I., appointed by Mr. Bush and reappointed by Mr. Obama. Known as a straight arrow, he led the bureau for 12 years, making him the longest-serving director other than J. Edgar Hoover.

“They’re taking the Clinton playbook from a much later stage in that investigation when gradually the public was itself questioning Ken Starr’s behavior and outraged by the Starr report,” said Ken Gormley, president of Duquesne University and author of “The Death of American Virtue,” a definitive history of the struggle between Mr. Clinton and Mr. Starr. “But what can happen when you start attacking someone who you haven’t seen do anything and they’re only doing their job, they may be generating significant sympathy for the special counsel and undercutting their purpose.”

Veterans of the Clinton team said that assailing Mr. Starr’s handling of the investigation was important in rallying Democrats behind Mr. Clinton and giving them a focus other than the president’s conduct.