WASHINGTON—The U.S. government estimates as many as six million households may have to pay a penalty for not having had health-insurance coverage last year as required under the Affordable Care Act, officials said Wednesday.

About 150 million taxpayers are expected to file returns during the coming tax season, said Mark Mazur, assistant secretary for tax policy at the Treasury Department. The tax-filing process this year is expected to be trickier because Americans will, in some cases, have to pay a penalty or get smaller...