"Here, the question is not whether a state prosecutor can indict a sitting President," the papers said. "Instead, the issue is whether a third party, having been duly served with a state grand jury subpoena seeking the books and records of a number of individuals and corporate entities, including Newport Cigarettes Shop those of the President, must comply with the subpoena."
Trump's lawyers have called the subpoena requests by Vance, a Democrat, a "bad faith effort to harass" Trump.
Trump's lawyers asked Marrero to declare the subpoena unenforceable until Trump leaves office.
The subpoena pertained to an investigation seeking records related to payments that former Trump lawyer Michael Cohen helped arrange to the porn actress Stormy Daniels and model Karen McDougal during the 2016 presidential campaign to keep either woman from speaking publicly about alleged affairs with Trump.
Trump has denied any sexual relationship with either woman, saying any payments were personal matters, not campaign expenses.
Cohen is serving a three-year prison sentence after pleading guilty to various federal charges, including Cheap Newport 100s Cigarettes Website campaign finance fraud and lying to Congress.
The case before Marrero comes as Trump's lawyers fight off other efforts to get his tax records.