QUITO, Ecuador—Ecuador has ordered 20 U.S. Defense Department employees attached to the U.S. Embassy in Quito to leave the country next week, U.S. officials confirmed, further straining an already rocky relationship between President Rafael Correa and the Obama administration.

The Ecuadorean government made the request on April 7, asking the U.S. embassy in Ecuador to end the activities of the Security Cooperation Office by Wednesday, said Jeffrey Weinshenker, an embassy spokesman. He said that the civilian and military workers...