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.