The U.S. Supreme Court on Thursday denied President-elect Donald J. Trump’s emergency bid to halt his criminal sentencing in New York, all but ensuring it would proceed as planned on Friday.

In a brief unsigned order, a five-justice majority noted that Mr. Trump was not facing jail time and that he could still challenge his conviction “in the ordinary course on appeal.”

Although Mr. Trump had argued that being sentenced 10 days before his inauguration would distract from the presidential transition, the majority held, “The burden that sentencing will impose on the president-elect’s responsibilities is relatively insubstantial.”

The majority included Chief Justice John G. Roberts Jr. and Justices Sonia Sotomayor; Amy Coney Barrett; Elena Kagan and Ketanji Brown Jackson. Four of the court’s conservative justices — Clarence Thomas, Samuel A. Alito Jr., Neil M. Gorsuch and Brett M. Kavanaugh — noted dissents without providing reasons.

The sentencing is now free to move forward on Friday morning in the same Lower Manhattan courtroom where Mr. Trump was convicted on 34 felony counts of falsifying records to cover up a sex scandal that had endangered his 2016 presidential campaign. The president-elect has indicated he plans to appear virtually.

After a series of unsuccessful legal maneuvers in New York State courts, the former and future president had hoped to prevail before a friendlier audience: a Supreme Court with a 6-to-3 conservative majority that includes three justices Mr. Trump appointed during his first term.

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