As Mr. Mnuchin’s second hearing of the day drifted past 5 p.m., he erupted in frustration at Representative Maxine Waters, who oversees the House Financial Services Committee, saying he needed to go so he could attend a meeting with an official from Bahrain.

When Ms. Waters refused to conclude the hearing and said he could leave if he wished, Mr. Mnuchin advised her to use her gavel and dismiss him.

“If you wish to keep me here so that I don’t have my important meeting and continue to grill me, then we can do that,” Mr. Mnuchin said. “I will cancel my meeting and I will not be back here. I will be very clear, if that’s the way you would like to have this relationship.”

Mr. Mnuchin and the I.R.S. have until the end of Wednesday to comply with Mr. Neal’s request, based on a deadline the lawmaker outlined last week. What comes next depends on their answer. The Treasury secretary told reporters after the hearing that it would be a “good guess” that his department would respond to the request by the deadline.

If the I.R.S. complies, Mr. Neal is unlikely to immediately say anything else to the public. If the agency objects to the request or does not answer, Mr. Neal could press his case and warn the agency that House Democrats will view not handing over the documents as a violation of the law.

At that point, Mr. Neal is likely to turn to the courts to try to enforce his request, teeing off a legal battle that could take months or years to sort out and could wind up at the Supreme Court.

Charles Rettig, the I.R.S. commissioner, under questioning by Democrats in another House hearing room, did his best to avoid divulging internal deliberations over Mr. Neal’s request, but said that he and the I.R.S. general counsel had been “interactive” with the Treasury Department in sorting out how to respond. He did not clearly state who he believed should have a say in the outcome.