Judge Sullivan ruled last month that the lawsuit, which was filed by more than 200 members of Congress, should be allowed to proceed to the evidence-gathering phase.

The complaint argues that Mr. Trump is accepting financial benefits from foreign governments in violation of the constitutional restrictions laid out in the so-called emoluments clauses. For example, it alleges that payments by foreign officials who book rooms at the Trump International Hotel in Washington fly in the face of the constitutional language designed to limit corrupt behavior by federal officials.

Judge Sullivan’s rulings so far have dovetailed with an earlier decision by Judge Peter J. Messitte of Federal District Court in Greenbelt, Md., who is overseeing a similar lawsuit filed by the State of Maryland and the District of Columbia.

The Maryland case is at least temporarily on hold. After Judge Messitte refused the department’s request for permission to appeal his rulings, the Justice Department sought an emergency review of his decisions by a three-judge panel of the United States Court of Appeals for the Fourth Circuit in Richmond, Va. Those judges agreed to stay the proceedings while they review Judge Messitte’s decisions. At an oral hearing, they appeared critical of those rulings and sympathetic to the president’s arguments.

Looming over both cases is the fact that Mr. Trump’s term is more than half over. Legal experts expect that the Supreme Court will eventually be asked to decide the constitutional questions, but the plaintiffs in both lawsuits hope that they will be allowed to gather evidence while Mr. Trump is still in office.