Donald Trump has vowed to appeal after the Supreme Court refused to delay his sentencing related to the hush money case, asserting that “Justice will prevail.

Prosecutors have been ordered to respond to the president-elect’s request by Thursday morning, suggesting the court could rule before Friday’s scheduled sentencing.

Days away from his criminal sentencing in New York, President-elect Donald J. Trump is seeking a late-stage rescue from the U.S. Supreme Court that would shut down the case before he returns to the White House.

In an emergency application late Tuesday, Mr. Trump’s lawyers urged the justices to intervene and halt the sentencing, which is scheduled for Friday, 10 days before the presidential inauguration. The filing came after a New York appeals court rejected the same request on Tuesday and sharply questioned the validity of his effort to stave off the sentencing.

Mr. Trump argues that he is entitled to full immunity from prosecution, and even sentencing, now that he is the president-elect. His lawyers have based that claim on a Supreme Court ruling last year that granted former presidents broad immunity for official acts.

“This court should enter an immediate stay of further proceedings in the New York trial court,” the application said, “to prevent grave injustice and harm to the institution of the presidency and the operations of the federal government.”

The Supreme Court directed prosecutors to respond to Mr. Trump’s application by Thursday morning, an indication that the justices may act before the scheduled sentencing. A spokeswoman for the Manhattan district attorney’s office, which prosecuted Mr. Trump on charges of falsifying records to cover up a sex scandal, declined to comment, saying only that the office would respond in court papers.

The filing capped a whirlwind stretch of legal wrangling for the former and future president, who is scrambling to avoid the embarrassing spectacle of a sentencing.

Although the trial judge, Juan M. Merchan, has signaled that he would spare Mr. Trump jail time, his sentencing would carry symbolic importance. It would formalize Mr. Trump’s status as a felon, making him the first president to hold that dubious designation.

Mr. Trump’s lawyers argued in the Supreme Court filing that officially becoming a felon might also complicate Mr. Trump’s presidential duties.

“The prospect of imposing sentence on President Trump just before he assumes office as the 47th president raises the specter of other possible restrictions on liberty, such as travel, reporting requirements, registration, probationary requirements, and others — all of which would be constitutionally intolerable under the doctrine of presidential immunity,” Mr. Trump’s lawyers wrote.

“Indeed, every adjudication of a felony conviction results in significant collateral consequences for the defendant, regardless of whether a term of imprisonment is imposed.”

The filing noted that Mr. Trump had also asked New York’s highest court, its Court of Appeals, for an emergency stay, acknowledging that litigants must ordinarily seek relief from lower courts before turning to the Supreme Court. Mr. Trump’s lawyers wrote that the state court was unlikely to act in time and so “filing applications in both courts appears to be the only viable option.”

A Supreme Court stay might scuttle Mr. Trump’s sentencing for good.

The window to sentence him is rapidly closing — once he returns to the White House, Mr. Trump cannot face prosecution — and he would be 82 after his second term. It is unclear whether the trial judge would still seek to impose Mr. Trump’s sentence four years later.

For now, Mr. Trump’s odds of success at the Supreme Court are unclear. The district attorney’s office is likely to argue that the court does not have jurisdiction yet, because Mr. Trump has not exhausted his appeals in state court.

But while some legal experts doubted the merits of his application, the justices have come to his rescue before. The Supreme Court’s immunity ruling in July effectively thwarted a federal criminal case against Mr. Trump for his effort to subvert the 2020 election results.

And even as other courts have grown hostile to Mr. Trump’s arguments in the New York case, he might find a friendlier reception at the Supreme Court, where the 6-to-3 conservative majority includes three justices he appointed during his first term. Five can grant a stay.

Michael Dorf, a law professor at Cornell, said Mr. Trump may well prevail.

“The application raises a novel question about whether presidential immunity extends to a president-elect, and, given the Supreme Court’s handling of Trump’s prior cases, has a fair chance of success,” Professor Dorf said.

But he added that Mr. Trump “does not in any real sense need the relief he seeks” because “there is no risk that the further proceedings would impinge in any way on Trump’s time.”

Over the years, Mr. Trump had mixed success at the court. That changed in the term that ended in July, which included three big victories for the former president.

In addition to the immunity ruling, the court allowed him to seek another term despite a constitutional provision barring insurrectionists from holding office, and cast doubt on two of the four charges against him in the since-scuttled federal prosecution over his efforts to subvert the 2020 election.

The new application will be an early test of the court’s independence in Mr. Trump’s second term, and its ruling may set the tone for what will surely be a fire hose of litigation arising from Mr. Trump’s plans to test the limits of presidential authority in, among many other areas, immigration and the economy.

Mr. Trump’s application was filed by two of his picks for top jobs in the Justice Department: Todd Blanche, Mr. Trump’s choice for deputy attorney general, and D. John Sauer, his selection for solicitor general.

They argued that Mr. Trump’s status as president-elect required special protections.

“President Trump is currently engaged in the most crucial and sensitive tasks of preparing to assume the executive power in less than two weeks, all of which are essential to the United States’ national security and vital interests,” the application said.

“Forcing President Trump to prepare for a criminal sentencing in a felony case while he is preparing to lead the free world as president of the United States in less than two weeks imposes an intolerable, unconstitutional burden on him that undermines these vital national interests.”

In May, a New York jury found Mr. Trump guilty on 34 felony counts of falsifying business records to hide a payoff to a porn star. Ever since, he has sought to unravel the verdict or at least postpone the sentencing, an effort that gained momentum in the wake of the Supreme Court’s immunity ruling.

Justice Merchan, the trial judge, recently rejected Mr. Trump’s bid to throw out the case based on that ruling. The judge concluded that because the ruling concerned a president’s “official acts,” it did not apply to the New York case, which centered on the “decidedly personal acts of falsifying business records.”

Even Chief Justice John G. Roberts Jr., who wrote the majority opinion, held that not every act a president takes is official, Justice Merchan noted.

Mr. Trump this week filed a civil action against Justice Merchan to unwind his ruling, and asked the state appellate court to pause his sentencing. But at a hearing Tuesday, a lawyer for the district attorney’s office emphasized that the case centered on “unofficial conduct,” a sex scandal before his presidency. Mr. Trump’s claim of immunity, the lawyer said, was “baseless.”