WASHINGTON—The Senate is expected to confirm Steven Mnuchin as the 77th Treasury secretary as soon as Monday night, which would end one of the most protracted confirmation battles for the position and place the 54-year-old financier in a leading position to advance President Donald Trump’s promises to refashion the U.S. tax code and financial regulation.

Mr. Mnuchin has faced opposition from Democrats for several reasons, including his role rehabilitating the failed IndyMac Bank, later rebranded OneWest Bank, by moving thousands...