President Donald Trump next week will make expansion of apprenticeship programs the center of his labor policy, aimed at filling a record level of open jobs and drawing back Americans who have left the workforce.

Apprenticeships are an underused form of workforce training in the U.S. compared with European countries and have a proven record of good outcomes for workers, administration officials said. Nine in 10 Americans who complete apprentice training land a job, and their average starting salary is $60,000 a year, according...