PRESCOTT, Ariz. - The parents of a 26-year-old American who Islamic State extremists said was killed in an airstrike in Syria said in a statement addressed to group leaders that the claim of their daughter's death concerned them but they were still hopeful she was alive.

The Islamic State group said on Friday that Kayla Jean Mueller of Prescott, Ariz., died in a Jordanian airstrike, but the government of Jordan dismissed the statement as propaganda. The U.S. said it had not seen any evidence to corroborate the report.

"Their nerves are absolutely frayed," family friend Todd Geiler said Saturday after leaving the Mueller home.

Mueller is the only known remaining U.S. hostage held by the Islamic State group.

If the death is confirmed, she would be the fourth American to die while being held by Islamic State militants. Journalists James Foley and Steven Sotloff and aid worker Peter Kassig were beheaded by the group.

"You told us that you treated Kayla as your guest, as your guest her safety and well-being remains your responsibility," Mueller's family said in a short statement released Friday.

The road leading to her family's home in Prescott remained blocked by authorities Saturday.

FBI agents also have been with the family and remain at their home, Yavapai County Sheriff's spokesman Dwight D'Evelyn said.

Mueller is an aid worker who previously volunteered with organizations in India, Israel and the Palestinian territories.

"The common thread of Kayla's life has been her quiet leadership and strong desire to serve others," Mueller's family said.

Her identity had not been disclosed until now out of fears for her safety. Her family said she was taken hostage by the Islamic State group on Aug. 4, 2013, while leaving a hospital in Syria.

"The secrecy issue was at the demand of her captors," Geiler said. "There were just a few of us around town who knew, for lack of better words, the living hell the family was going through."

He said the family planned to release another statement sometime Saturday. However, a family spokesman said there were no plans to do so.

"Kayla was a wonderful young lady," Geiler said. "She was not one to sit in the back and take a back seat on issues, and she truly thought she could make a difference."

Jordan has been launching airstrikes against the extremist group in response to a video released this week that shows Jordanian pilot Lt. Muath al-Kaseasbeh being burned to death in a cage.

Al-Kaseasbeh, whose F-16 came down in December while conducting airstrikes as part of a campaign against the militants by a U.S.-led coalition, was believed to have been killed in January.

The Islamic State group said in a statement that Mueller was killed in the militants' stronghold of Raqqa in northern Syria during midday prayers in airstrikes that targeted "the same location for more than an hour."

It published photos purportedly of the bombed site, showing a severely damaged three-story building, but offered no proof or images of Mueller.

The statement said no Islamic State militants were killed in the airstrikes, raising further questions about the veracity of the claim.

The Jordanian government said it was investigating, and American officials have said they are looking into the report. Both governments are skeptical of the claim.

A U.S. official said coalition aircraft conducted bombing near Raqqa on Friday, but he had nothing to confirm the claim that the American captive was killed in the airstrike. He spoke on condition of anonymity because he wasn't authorized to discuss the issue with reporters.

Mueller had been working in Turkey assisting Syrian refugees, according to a 2013 article in The Daily Courier, her hometown newspaper.

According to the newspaper, Mueller had been working with the humanitarian aid agency Support to Life, as well as a local organization that helped female Syrian refugees develop skills.

The Jordanian government said dozens of its fighter jets had bombed Islamic State training centers and weapons storage sites. Activists who monitor the Syrian conflict from inside the country said airstrikes resumed Saturday.