A human-rights commission reporting to Saudi King Salman is investigating the alleged torture of detained women’s rights activists, including accusations of waterboarding and electrocution, according to government officials and other people familiar with the activists’ situation.

A top aide to Crown Prince Mohammed bin Salman, Saud al-Qahtani, allegedly oversaw some aspects of the torture and threatened at least one jailed woman with rape and death, according to testimony before the commission, those officials and others...