WASHINGTON — After the Islamic State declared that Kayla Mueller, a 26-year-old humanitarian worker from Arizona it was holding hostage, was killed in an airstrike in Syria in 2015, her parents became increasingly frustrated with the government’s slow hunt for answers about her death and turned to a former F.B.I. agent for help.

Now, he has unearthed a clue: Ms. Mueller may have been executed on the orders of Abu Bakr al-Baghdadi, the slain Islamic State leader, not in a Jordanian airstrike as the Islamic State long claimed.

ISIS operatives killed Ms. Mueller because she knew the identities of Mr. al-Baghdadi and of Abu Muhammad al-Adnani, the group’s spokesman at the time who was believed to be the second-in-command, and could be a security risk for them, according to Umm Sayyaf, the wife of a close al-Baghdadi associate. She was interviewed by the former F.B.I. agent working with the Mueller family, Ali Soufan.

Ms. Mueller, whom Mr. al-Baghdadi raped repeatedly, was moved from location to location in the final months of her life, Umm Sayyaf said. At one point, Ms. Mueller was moved because the oldest of Mr. al-Baghdadi’s multiple wives assaulted Ms. Mueller, breaking a watch that Mr. al-Baghdadi had given Ms. Mueller as a Ramadan gift and kicking her out of the home.