WASHINGTON—President Donald Trump will nominate Charles Rettig, a California tax lawyer, to run the Internal Revenue Service, a person familiar with the matter said Monday.

If confirmed by the Senate, Mr. Rettig will take one of the most thankless jobs in Washington. He would run an agency that despite a shrunken budget is responsible for implementing the sprawling Republican overhaul of the U.S. tax system passed by Congress last year. He would also be charged with overseeing the agency Mr. Trump has said has been auditing...