As he awaits his fate, any changes to Mr. Menendez’s political future have the potential to reverberate far beyond New Jersey. With Republicans holding a slim majority in the Senate, much of President Trump’s legislative agenda has stalled or failed. If Mr. Menendez were to be exonerated, or if there were a mistrial, and he decided to run for re-election, he could face an uphill battle after weeks of politically damaging testimony. But the senior senator from New Jersey has shown no willingness to step down.

The note from jurors came three hours after they had started a new round of deliberations after the juror was excused on Thursday so she could go on a vacation that had been planned before the trial began. Her excusal, and subsequent media interviews, injected drama into a case that had been relatively staid.

“I’m praying for him that they do not find him guilty,” Evelyn Maultsby, the excused juror, told reporters on Thursday.

Ms. Maultsby’s comments, including accusing fellow jurors of preventing her from expressing her opinion, were widely broadcast, bringing national attention to the trial.

On Monday morning, four jurors admitted in court to having seen news reports about Ms. Maultsby’s comments, even though they had been warned by Judge Walls against following any trial coverage.

Mr. Lowell said Ms. Maultsby’s complaint that she was prevented from giving a note to the judge, represented the “potential that there was misconduct.”

But Judge Walls was emphatic that the trial would proceed. “If you were suggesting that somehow this is a genesis of a mistrial, I am saying that you’re off target,” he said. “You’re way off target. There is no way in God’s green earth that this judge is going to grant a mistrial when we have available to us a new jury.”