Mr. Trump’s Treasury Department was facing a midnight deadline to respond to a letter sent last week by Mr. Neal, who issued a formal request for six years of Mr. Trump’s personal and business tax records.

White House officials have called the request a political fishing expedition, and Mr. Mnuchin has said that furnishing the returns could open the floodgates to weaponizing the Internal Revenue Service.

Mr. Mnuchin signaled his intention on Tuesday, when he told members of Congress that the president had no legal obligation to make his tax returns public. While Mr. Mnuchin insisted that he would follow the law, he also defended Mr. Trump’s right to keep his tax returns private.

Mr. Mnuchin reiterated his position that the privacy of taxpayers, including that of the president, was paramount during a moderated discussion on Wednesday at the International Monetary Fund.

“We are going to respond carefully. We need to make sure that the I.R.S. and individual taxpayer information does not just become subject to political winds,” Mr. Mnuchin said, adding that he had been consulting with the Justice Department on the matter. “Whether it is this party or a different party over time, I take the obligation very seriously to make sure that we follow the law correctly.”

But the provision in tax law that Democrats are using appears to give the Trump administration little leeway to resist their request; it says merely that the Treasury secretary “shall” furnish the requested information. It was used several times by House Republicans when they were investigating whether the I.R.S. had improperly delayed applications by conservative groups for nonprofit status.

Some Democrats quickly denounced the Treasury Department’s decision on Wednesday.

“How many lawyers and how much time does it take for Secretary Mnuchin to understand that ‘shall’ means ‘shall’?” said Representative Lloyd Doggett of Texas, a member of the Ways and Means Committee. “Again, believing that he is above the law, Trump is engaged in obstruction.”