WASHINGTON—An American woman who died while in Syrian government custody is believed by aid groups and activists to have been executed, spurring concern for other U.S. citizens who may be detained by the regime of President Bashar al-Assad.

The family of Layla Shweikani, a 26-year-old Illinois native who traveled to Syria to provide humanitarian relief to civilians affected by the country’s civil war, was notified last month by the Syrian government that she died in December 2016, supporters said. She had been detained by...