Without naming Mr. Trump during Thursday’s sentencing hearing, Judge Jackson cited “entirely inappropriate” comments by partisan onlookers. She amplified her warning on Tuesday, saying, “I need to state this clearly, that any attempt to invade the privacy of the jury is completely antithetical to our entire system of justice.”

The case also exposed a rift between career prosecutors in the United States attorney’s office for the District of Columbia and their superiors at Justice Department headquarters. Four prosecutors withdrew from the case this month after Mr. Barr overruled their sentencing recommendation and insisted the department request a less severe prison term than the seven to nine years the prosecutors sought.

Tuesday’s four-hour hearing was itself highly unusual. Last-ditch attempts by defense teams to keep their defendants out of prison are typical, but their motions rarely receive so much judicial attention. Legal experts said that Judge Jackson is probably both anticipating Mr. Stone’s appeal of his conviction and striving to defend the integrity of the proceedings in the face of attacks by the president and his allies.

Three months after the jurors rendered their verdict, she summoned about a dozen of them back to the courthouse. Three of them, including the forewoman, were questioned on the stand, as was one of the prosecutors who withdrew from the case.

Defense lawyers claim that the forewoman failed to disclose important information during the jury selection process that would have prompted them to move to exclude her from the panel. Tomeka Hart, a program officer for the Bill and Melinda Gates Foundation, identified herself as the forewoman in a Facebook post this month.

She voiced concerns in that post about Mr. Barr’s decision to overrule the prosecutors, saying: “It pains me to see the DOJ now interfere with the hard work of the prosecutors. They acted with the utmost intelligence, integrity and respect for our system of justice.”

On Tuesday, Mr. Stone’s lawyers claimed that during jury selection, the forewoman concealed her level of knowledge about Mr. Stone and his relationship with the president, as well as animosity toward Mr. Trump. They said they only recently learned of her views by examining her posts on social media, even though they had hired consultants to help them identify potential jurors who could be biased.