WASHINGTON—The Treasury Department didn’t plan to meet Wednesday’s deadline for handing President Trump’s tax returns to House Democrats, as it continued to review the request, Secretary Steven Mnuchin wrote.

Mr. Mnuchin, in a letter on Wednesday to House Ways and Means Committee Chairman Rep. Richard Neal (D., Mass.), said the request raises “serious issues” about whether the committee has a legitimate legislative purpose and how broad congressional investigatory powers are under the Constitution. Those are the issues that...