The parents of an American aid worker who is being held hostage by Islamic State (ISIS) militants are keeping the faith that their daughter is still alive amid reports that she perished in a Jordanian airstrike against the group in Syria.

In their statement, the parents of Kayla Jean Mueller appealed to their daughtor's captors to keep her safe and told them that her wellbeing is their responsibility.

"To those in positions of responsibility for holding Kayla; in adherence to your warnings and out of concern for Kayla's safety, we have been silent until now," the statement said. "This news leaves us concerned, yet, we are still hopeful that Kayla is alive. We have sent you a private message and ask that you respond to us privately.

"You told us that you treated Kayla as your guest, as your guest her safety and well-being remains your responsibility," they added.

Before she travelled to Syria to help children affected by armed conflict, Mueller volunteered for humanitarian missions in India, Israel and Palestine.

"Syrians are dying by the thousands, and they're fighting just to talk about the rights we have," Mueller said, while addressing the Prescott Kiwanis Club in May, 2013. "For as long as I live, I will not let this suffering be normal. (I will not let this be) something we just accept."

The group has claimed that the American became the casualty of the airstrike ordered by Jordanian King Abdullah, in retaliation for the ISIS's uploaded video of Jordanian pilot Muadh al-Kasasbeh being burned alive inside a cage.

The King reportedly cut short his US visit to go on air on state television to issue a warning against members of the ISIS, stating that they would launch a relentless war against the group not only to avenge the brutal slaying of the pilot but also to "protect their faith, values and human principles."

Bernadette Meehan, a spokesperson for the National Security Council, said that while the United States government is very concerned about the reports, the White House has not seen hard evidence that corroborates ISIS's claim.

Meanwhile, Jordanian officials are not convinced that the American died as a result of the airstrike, citing the group's dishonesty in their previous dealings.

"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," Mohammed al-Momani, Jordanian government spokesperson, said.