More specifically, Judiciary Committee members will press Mr. Mueller to expand on a handful of the dozen or so episodes of potential obstruction cited in his report — including Mr. Trump ordering Don McGahn, the White House counsel at the time, to fire Mr. Mueller, then later instructing Mr. McGahn to deny any such effort. The Trump administration has blocked Mr. McGahn himself, along with many others named in the report, from testifying before Congress.

Complicating matters, Mr. Mueller is seen as, if not a hostile witness, at least a grudging one. In his brief public statement on his report in May, he all but begged Congress not to force him to testify and warned that, if called, he would provide no information beyond the scope of his report. “The report is my testimony,” he asserted.

Lawmakers have held mock sessions, running through practice questions and fine-tuning them to be as pointed — and un-duckable — as possible. On “Fox News Sunday” this weekend, Jerrold Nadler, the Judiciary Committee chairman, offered an example of how this could go: “‘Look at Page 344, Paragraph 2. Please read it. Does that describe obstruction of justice? Did you find that the president did that?’”

That said, Democrats acknowledge that it will be a challenge to get anything new out of Mr. Mueller. As Mr. Schiff recently joked, “We have a far better chance of the love affair in North Korea working out” than of getting Mr. Mueller to say whether Mr. Trump should be indicted after leaving office.

With this in mind, Mr. Schiff noted that he and his fellow Democrats “have to decide how much of our time we want to spend fighting with him to discuss things outside the report.”

The House Democratic leadership has warned members against grandstanding.

Republicans also will be aiming to keep their team from overreaching and appearing too partisan in their attacks on Mr. Mueller, who is widely respected. It bears recalling that Representative Devin Nunes of California, the top Republican on the Intelligence Committee, has been among Congress’s most fervent crusaders in trying to discredit the Mueller investigation. Representative Louie Gohmert of Texas, a Republican who sits on the Judiciary Committee, put out a report titled “Mueller Unmasked” that claimed to expose “the disreputable, twisted history” of the special counsel.

Looking to stir up trouble, Mr. Trump has waded into the drama. On Monday, he tweeted out fresh insults of Mr. Mueller and warned that the hearings would end badly for Democrats. He even found a way to work in Hillary Clinton, claiming, bizarrely, that all of her people had been “given immunity.”