And Senator Tom Carper of Delaware, the top Democrat on the Homeland Security and Governmental Affairs Committee, sent letters to the top ethics officials at 17 government agencies, asking if they had been in touch with officials of the Trump transition, whether they had received financial disclosure statements, and whether any Trump pick “refused to provide any information that you believe is necessary to conduct a conflicts analysis as required by law.”

“Given the large and complex financial holdings and boundless, serious potential for conflicts of interest,” Senator Chuck Schumer, Democrat of New York, said in an email, “these nominees need to turn over all relevant financial and background information so that senators can thoroughly review their record before going forward with any hearings.”

Several of Mr. Trump’s cabinet picks would be among the wealthiest public servants in modern history. That alone presents a significant financial-vetting challenge to Senate Republicans, who hope to begin confirmation hearings in a few weeks. Mr. Trump’s selection process — begun, unlike that of most predecessors, after his election rather than before — may have added to the challenge of moving quickly now.

“They need to step on the gas and get it done,” said Richard Painter, a law professor at the University of Minnesota who served as chief ethics counsel to President George W. Bush. “They need to tell the Senate what they are going to do with their assets.”