Donald J. Trump’s lawyers have tried just about everything to overturn his criminal conviction in New York.

They invoked the fine print of a recent Supreme Court ruling on presidential immunity. They argued the case would obstruct “the overwhelming national mandate granted to him” on Election Day. They even tried to move it to federal court, accusing the state court judge who presided over the trial of bias.

Now, Mr. Trump’s lawyers are taking a new tack: undermining the jury that convicted him.

In a letter to the judge that was unsealed on Tuesday, Mr. Trump’s lawyers claimed to have uncovered “evidence of grave juror misconduct during the trial,” in which he was convicted of falsifying records to cover up a sex scandal. “The jury in this case was not anywhere near fair and impartial,” Mr. Trump’s lawyers asserted.

But the nature of any misconduct that they are claiming occurred is unclear. The letter and several related replies from prosecutors — all of which were exchanged behind the scenes in recent days — are heavily redacted. And the identify of the purported whistle-blower who raised the accusation is secret.

In a statement, a spokesman for Mr. Trump implied that a juror may have expressed some political leanings during the trial, contending that “partisan political motivations infected nearly every aspect of this witch hunt, including the jury room.”

But like Mr. Trump’s other maneuvers to throw out his 34 felony-count conviction, this too, is a long shot.