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Donald J. Trump began an effort on Monday to throw out his recent criminal conviction in Manhattan and postpone his upcoming sentencing, citing a new Supreme Court ruling that granted him broad immunity from prosecution for official actions he took as president, according to a person with knowledge of the matter.

In a letter to the judge overseeing the case, Mr. Trump’s lawyers sought permission to file a motion to set aside the verdict, doing so just hours after the Supreme Court issued its landmark ruling involving one of Mr. Trump’s other criminal cases. The letter will not be public until Tuesday at the earliest, after which prosecutors will have a chance to respond.

The move from Mr. Trump’s lawyers came 10 days before the judge was set to sentence the former president for his crimes in Manhattan, where a jury convicted him on 34 felony counts related to his cover-up of a sex scandal in the run-up to the 2016 election. Mr. Trump’s lawyers asked the judge, Juan M. Merchan, to postpone the July 11 sentencing while the judge weighs whether the Supreme Court ruling affects the conviction.

The effort to set aside the conviction might be a long shot. The Manhattan case centers on acts Mr. Trump took as a candidate, not a president.

Yet his lawyers are likely to argue that prosecutors built their case partly on evidence from his time in the White House. And under the Supreme Court’s new ruling, prosecutors not only may not charge a president for any official acts, but also cannot cite evidence involving official acts to bolster other accusations.

It is unclear how the Manhattan district attorney’s office, which brought the case, will respond, or whether the judge will delay the first sentencing of an American president. But Mr. Trump’s effort appeared to cause at least a brief interruption: The district attorney’s office did not on Monday make a sentencing recommendation to the judge about whether to imprison Mr. Trump, as was expected.

A spokeswoman for the district attorney’s office declined to comment.

It might be too late for Justice Merchan to revisit the conviction. The deadline for filing post-trial motions was last month, and it is unclear whether the judge will seriously entertain the motion, even in light of the high court’s decision. Instead, he might direct Mr. Trump’s lawyers to raise the issue when they appeal the conviction after he is sentenced.

The sentencing is likely to be the only moment of criminal accountability that the four-times indicted former president will face before Election Day, when he hopes to reclaim the White House. Mr. Trump faces up to four years in prison, but could receive probation on the convictions for falsifying business records — among the lowest level of felonies.

Mr. Trump’s other criminal cases are mired in delay, and the Supreme Court ruling on Monday will almost certainly postpone his trial in Washington, where he is accused of plotting to subvert the 2020 election.

In the Manhattan case, the district attorney, Alvin L. Bragg, will eventually submit his recommendation to Justice Merchan, but it is unclear whether those legal papers will be public.

Unlike in federal court, sentencing papers in New York State courts are generally confidential unless the judge authorizes their release. That means the world may not learn of Mr. Bragg’s recommendation until Justice Merchan sentences Mr. Trump, as he is expected to do just days before Mr. Trump is to be formally nominated for president at the Republican National Convention.

The judge faces an unprecedented conundrum with equally unprecedented legal and political ramifications. His decision is sure to alienate broad swaths of the country no matter what it is.

Imprisoning Mr. Trump could exacerbate the nation’s political divisions. Yet if Justice Merchan spares him, it could give the impression that the former president received special treatment. Mr. Trump’s front-runner status in the 2024 presidential campaign further compounds the judge’s dilemma, raising the prospect that his decision could shape the race and its outcome.

The judge, a former prosecutor who occasionally clashed with Mr. Trump and his lawyers, has several options at his disposal. Mr. Trump could face a few months in jail or several years in prison. But it’s just as likely that he won’t ever see the inside of a cell: There is no requirement that the judge impose time behind bars, and Justice Merchan could sentence him to home confinement or probation. He could also postpone any sentence until after the election, or, after Mr. Trump serves his second term in office, should he be re-elected. A sitting president cannot be required to serve time.

After Mr. Bragg submits his recommendation, Mr. Trump’s lawyers are entitled to respond with their own recommendation. It is unclear whether that submission will be confidential as well.

The judge could draw on feedback from the New York City probation department, which met with Mr. Trump last month and will make its own confidential recommendation for his sentence.

Several factors are working in Mr. Trump’s favor: A major one is that he is a first-time felon convicted of a nonviolent crime. On the other hand, Mr. Trump’s behavior before and during the trial could cost him his freedom: He has shown no remorse and on 10 separate occasions violated a gag order that prohibited him from attacking jurors, prosecutors and witnesses — including the prosecution’s star witness, Mr. Trump’s former fixer, Michael D. Cohen.