A mid-level state appeals court last week reinstated the gag orders, which barred Trump and his lawyers from making public statements about court staff. An appeals court judge had temporarily paused the order on November 16.
Justice Arthur Engoron imposed the gag order on Trump on October 3 after Trump accused Engoron's top clerk of political bias in a post on his Truth Social platform. The post left the court "inundated" with hundreds of threats from Trump supporters, Engoron said in a court filing.
Engoron later restricted Trump's lawyers statements about his staff.
In Monday's court filing, Trump lawyer Clifford Robert asked the mid-level appeals court, known as the Appellate Division, to allow Trump to appeal its reinstatement of the orders to the Albany-based Court of Appeals.
Robert said the orders violated the US and New York state Constitutions by restricting Trump's "First Amendment right to highlight serious concerns raised by the public and partisan activities of Justice Engoron's Principal Law Clerk during an ongoing bench trial."
A spokesperson for the New York Attorney-General's office, which brought the case against Trump and his family real estate company, declined to comment.
Trump, the frontrunner for the 2024 Republican presidential nomination, is accused in the case of inflating his net worth by billions of dollars to dupe lenders and insurers. The trial largely concerns damages, because Engoron has already found that Trump's financial statements were fraudulent.
New York Attorney-General Letitia James is seeking $US250 million ($A376 million) in penalties, and wants Trump banned from New York state real estate business.
Trump has denied wrongdoing and said James, an elected Democrat, is politically biased against him.
Engoron has already fined Trump $US15,000 ($A22,500) for twice violating the order and warned that future breaches would be met with steeper penalties, including imprisonment.
Trump is under a similar gag order in an unrelated criminal case over his efforts to change the results of the 2020 election.
He faces four federal and state criminal indictments, including two over his efforts to overturn his 2020 loss, and has pleaded not guilty in all of those cases
None of them have dented his commanding lead in the race for the Republican nomination to challenge Democratic President Joe Biden in the November 2024 election.
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