IS said Kayla Mueller, 26, was killed on Friday in Raqqa, Syria, by Jordanian fighter jets, but provided no proof.
In a statement released on Friday night, Carl and Marsha Mueller, who live in Arizona, pleaded with IS to contact the family privately.
“You told us that you treated Kayla as your guest, as your guest her safety and well-being remains your responsibility,” they said in a message directed to “those in positions of responsibility for holding Kayla.
“Kayla’s mother and I have been doing everything we can to get her released safely.”
Jordan dismissed the claim as a “PR stunt” intended to divide the international coalition after IS brutally and publicly murdered one of their pilots, Flight Lieutenant Muath al-Kaseasbeh, whose plane was shot down over Raqqa, an IS stronghold.
US intelligence was attempting to verify the report amid fears that IS may have murdered Mueller and was trying to pin the blame on an American ally in the Arab world.
Mueller was kidnapped in August 2013 while in Aleppo, northern Syria. She is said to have accompanied her Syrian boyfriend on a job he was doing at the compound of the Spanish branch of Médecins Sans Frontières. It said it had not known she would be coming and she was never employed by MSF in any capacity.
IS had previously demanded a several million dollar ransom for her release. The group reportedly put out a proof-of-life video last year, showing Mueller in a hijab pleading for her life.
It emerged on Friday that the Muellers had rejected the idea of launching a risky mission to rescue her, and asked that her release be negotiated. If her death is confirmed, she would be the fourth American and the first US woman killed while held captive by the jihadist group.
IS has previously lied about the state of its hostages and it spent weeks engaging in negotiations over the Jordanian pilot only for it to emerge it had in fact killed him last month. Jordan responded by executing two terrorists – including a female suicide bomber.