EL PASO, Texas—The Trump administration decision to keep hundreds of U.S. troops based in Syria was driven by allies who said they wouldn’t stay behind to prevent a resurgence of the Islamic State extremist group without an American presence, a senior U.S. defense official said Saturday.

Faced with the likelihood that French and British forces would leave following a U.S. withdrawal, the administration agreed last week to maintain a residual force. Any American presence would mean the U.S. military would need to keep control...