President Donald Trump is giving businesses expanded authority to design their own apprenticeship programs in an executive order signed June 15.

The Labor Department’s current system of funding apprenticeships after they’ve met longstanding quality standards is “very prescriptive and very restrictive” and “really fails the American worker,” a senior White House official said on a call with reporters.

The executive order is intended to improve industry flexibility and scale up the earn-as-you-learn job training program by allowing companies to tailor...