In a letter of his own, Representative Jerrold Nadler, Democrat of New York and the chairman of the Judiciary Committee, raised what could be a more troubling outcome for the president. He said his committee was considering potential violations of the ban on profiting from the presidency as part of its impeachment investigation.

“Potential violations of the foreign and domestic emoluments clauses of the Constitution are of grave concern to the committee as it considers whether to recommend articles of impeachment,” he wrote. The letter was also signed by Representative Steve Cohen, Democrat of Tennessee, who leads a relevant subcommittee.

The White House did not immediately comment on the letters.

Mr. Trump and Mr. Pence drew sharp criticism this week after the vice president and his coterie of family, aides and security stayed a night at the Trump International Golf Links & Hotel in Doonbeg during a trip to Ireland, despite the fact that it was 181 miles from his meetings in Dublin. Government receipts for such stays usually run in the tens of thousands of dollars — or far more, depending on the size of the group and the length of stay.

Mr. Pence’s chief of staff initially told reporters that the vice president, who has family roots in Doonbeg, had chosen the accommodation at the “suggestion” of Mr. Trump. Facing a flurry of criticism for the choice, the vice president’s office later released a statement saying that “at no time did the president direct our office to stay at his Doonbeg resort.”

Mr. Cummings’s request asked for documents showing itemized expenses from the most recent Ireland trip, any communications related to Mr. Pence’s accommodations, and records related to Mr. Trump’s own stay at the Doonbeg resort during an official visit to Ireland in June.