Last week, a unanimous three-judge panel of a federal appeals court in Manhattan ruled against Mr. Trump. The court, in a focused ruling, said state prosecutors may require third parties to turn over a sitting president’s financial records for use in a grand jury investigation.

Mr. Trump has fought vigorously to shield his financial records, and prosecutors in Manhattan have agreed not to seek the tax returns until the case is resolved by the Supreme Court. In exchange, they insisted on a very quick briefing schedule, one that would allow the court to announce whether it would hear the case as soon as next month and to issue a decision by June, as the presidential election enters its final stages.

Other cases involving Mr. Trump are also in the pipeline. They involve matters as diverse as demands from House Democrats for tax and business records, a request for access to redacted portions of the report prepared by Robert S. Mueller III, the special counsel, and challenges to Mr. Trump’s business arrangements under the Constitution’s emoluments clauses.

On Wednesday, the full United States Court of Appeals for the District of Columbia Circuit refused to rehear a ruling from a divided three-judge panel that Mr. Trump’s accounting firm must comply with the House Oversight and Reform Committee’s demands for eight years of his financial records. A lawyer for Mr. Trump said he would appeal that ruling to the Supreme Court, too.

The legal fight in the New York case began in late August after prosecutors in the office of the Manhattan district attorney, Cyrus R. Vance Jr., a Democrat, subpoenaed Mr. Trump’s accounting firm, Mazars USA, for his tax returns and those of his family business dating to 2011.

In Thursday’s filing, Mr. Trump’s lawyers called that move unprecedented.

“For the first time in our nation’s history,” they wrote, “a state or local prosecutor has launched a criminal investigation of the president of the United States and subjected him to coercive criminal process.”

The petition said the subpoena was “politically motivated” and sought information that was not relevant to any legitimate criminal inquiry.