“The only goal in this litigation,” prosecutors with the district attorney’s office wrote, “is to obtain as much delay as possible, through litigation, stays and appeals.”

A lawyer for Mr. Trump did not respond to a request for comment.

Mr. Trump’s lawsuit argued that the Constitution bars any investigation of a sitting president, especially by a local prosecutor. The argument has never been tested in court and could set a sweeping new precedent if the president is successful.

Lawyers for the president also have called the investigation by the Manhattan district attorney, Cyrus R. Vance Jr., politically motivated. Mr. Vance is a Democrat.

Mr. Vance’s office was responding to a letter filed on Wednesday by the Justice Department supporting a request from Mr. Trump, a Republican, that the judge in the case temporarily block the subpoena. Lawyers for the Justice Department, which is led by Attorney General William P. Barr, said the president had raised “significant constitutional issues” that should first be carefully considered by the court.

The Justice Department did not offer an opinion about the merits of Mr. Trump’s argument.

But the letter from Mr. Vance’s office indicated that the state prosecutors interpreted the Justice Department’s involvement as support for the president’s “extravagant claim that, given his current position, he and all of his prior business associates and related companies are immune, not just from prosecution, but from any routine grand jury inquiry into transactions undertaken before he was a government employee.”