No one knows the answer because in 230 years, no court has interpreted the clause. Nor has any president in modern history put it to the test as Mr. Trump has. Every other president since Jimmy Carter has gone to great lengths to avoid such entanglements, typically by putting their personal assets in independently managed blind trusts.

The debate now also engulfs Mr. Trump’s daughter Ivanka, who late last month became an unpaid assistant to the president. She, too, is covered by the emoluments clause, according to previous legal opinions issued by the Justice Department. Her company has 37 trademark applications pending in 10 countries, covering the sale of leather goods in China, jewelry in the Philippines and beauty products in Indonesia, The New York Times review shows.

The president’s critics argue that the emoluments clause prohibits Mr. Trump from accepting any economic benefit from a foreign power, just as it prevented Abraham Lincoln from accepting a gift of elephant tusks from the king of Siam in 1862. Otherwise, they argue, foreign governments could seek the president’s favor through actions like trademark registrations or pressure him by withholding approvals.

“If you could bring the founders back, to a person they would say, no, you can’t accept that,” said Norman L. Eisen, a co-founder of Citizens for Responsibility and Ethics in Washington, a liberal nonprofit group that filed a lawsuit in United States District Court against the president over emoluments. “How can we be confident that he is making decisions in the interest of the United States when he has these enormous potential inducements?”

But in a brief expected to be filed this month, Justice Department lawyers will counter that the framers of the Constitution meant only to rule out gifts and compensation for services, not ordinary, arm’s-length commercial transactions with foreign governments. Otherwise, they argue, the framers would have had to confront the potential effect of the ban on the nation’s earliest presidents, including George Washington, who supplemented his meager presidential salary partly by exporting flour and cornmeal to England and elsewhere.