WASHINGTON—President Trump said Wednesday he is looking at five potential candidates to replace his ousted national security adviser, John Bolton, and wished Mr. Bolton well—but only after slamming his views on a range of foreign-policy issues.

Mr. Bolton, a Republican foreign policy hawk who served as Mr. Trump’s third national security adviser, left his position a day earlier in the culmination of a range of disputes with the president, including differences over whether to engage in talks with Iran’s president or with the...