WASHINGTON—President Trump reignited his global trade war Monday, renewing his threat to hit partners with punishing tariffs even as he announced a three-week extension to negotiate deals.

Trump signed an executive order extending the date when his so-called reciprocal tariffs would take effect, with a pause previously scheduled to expire at 12:01 a.m. Wednesday. Additionally, Trump sent letters to a handful of nations outlining tariff rates they would pay if they didn’t strike trade deals with the U.S. by Aug. 1.