The parents of a 26-year-old woman who Islamic State extremists say has been killed in an airstrike in northern Syria have 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.

Kayla Mueller, whose identity had previously not been revealed at the request of the U.S. government and her family as the Pentagon worked to secure her, is believed to be the last American hostage the terrorist army is holding, since the beheading of Peter Kassig, a 26-year-old American aid worker, killed in November. In its statement, ISIS also claimed Mueller was the only person killed in the airstrike, claiming none of its fighters injured or killed.

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

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

In this May 30, 2013, photo, Kayla Mueller is shown after speaking to a group in Prescott, Ariz. A statement that appeared on a militant website commonly used by the Islamic State group claimed that Mueller was killed in a Jordanian airstrike on Friday, Feb. 6, 2015, on the outskirts of the northern Syrian city of Raqqa, the militant group's main stronghold. The IS statement could not be independently verified. (AP Photo/The Daily Courier, Matt Hinshaw) MANDATORY CREDIT (AP/The Daily Courier)

"You told us that you treated Kayla as your guest, as your guest her safety and well-being remains your responsibility," the statement from Mueller's parents says.

'You told us that you treated Kayla as your guest, as your guest her safety and well-being remains your responsibility'
— family statement

Mueller is an aid worker who has previously volunteered with aid groups in India, Israel and the Palestinian territories. 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 common thread of Kayla's life has been her quiet leadership and strong desire to serve others," Mueller's family said in the short statement released Friday.

Jordan has been launching airstrikes against the extremist group in response to a video released this week that shows captive 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 early January.

Friday's statement from the group said Mueller was killed in the militants' stronghold of Raqqa in northern Syria during Muslim prayers -- which usually take place around midday -- 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.

Jordanian government spokesman Mohammed al-Momani said it was investigating.

"But as a first reaction, we think it's illogical and we are highly skeptical about it. How could they identify a Jordanian warplane ... in the sky? What was the American lady doing in a weapons warehouse?" al-Momani said.

"It's part of their criminal propaganda. They have lied that our pilot is alive and tried to negotiate, claiming he is alive while they had killed him weeks before," he added.

American officials said they also were looking into the report.

“We are obviously deeply concerned by these reports," said National Security Council spokeswoman Bernadette Meehan. "We have not at this time seen any evidence that corroborates ISIL’s claim."

Mueller had been working in Turkey assisting Syrian refugees, according to a 2013 article in The Daily Courier, her hometown newspaper. She told the paper that she was drawn to help with the situation in Syria.

"For as long as I live, I will not let this suffering be normal," she said. "It's important to stop and realize what we have, why we have it and how privileged we are. And from that place, start caring and get a lot done."

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 Syrian government said Thursday that dozens of Jordanian fighter jets had bombed Islamic State training centers and weapons storage sites. It did not say where the attacks occurred.

The Jordanian military said its warplanes carried out a series of attacks Friday and "destroyed the targets and returned safely." It did not elaborate.

Activists who monitor the Syrian conflict from inside the country said coalition planes hit several targets on the edges and outskirts of Raqqa in quick succession.

A Raqqa-based collective of anti-IS activists known as "Raqqa is Being Slaughtered Silently" said the planes targeted multiple IS positions and headquarters in the western and eastern countryside of Raqqa, sending up columns of smoke. Explosions could be heard in the city. The collective said there were no recorded civilian casualties, and did not mention any IS casualties.

The Britain-based Syrian Observatory for Human Rights said dozens of IS members were killed in coalition airstrikes that targeted a tank and vehicle depot in the area of al-Madajen and at least six other IS positions, including a training camp and a prison.

The Associated Press contributed to this report