WASHINGTON—The Trump administration is considering major reductions in cultural exchange programs, including those for au pairs and summer workers, that allow young people from foreign countries to work in the U.S., people familiar with the administration’s planning said.

President Donald Trump’s “Buy American and Hire American” executive order, issued in April, calls for a review of U.S. immigration rules to ensure that the interests of domestic workers are protected. No decisions have been made, but supporters of the program...