The IS jihadists said Kayla Jean Mueller had been buried under rubble when a Jordanian warplane hit a building in the Syrian city of Raqa, their self-proclaimed "capital".
But the United States said there was no proof that the 26-year-old aid worker from Arizona had been killed.
Her parents appealed to her captors to contact them and for her safe return, in a statement carried by NBC News.
"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," said Carl and Marsha Mueller.
"You told us that you treated Kayla as your guest, as your guest her safety and wellbeing remains your responsibility," they said.
Jordan -- still reeling from the brutal murder of one of its pilots by IS -- rejected the jihadists' claim that its warplanes killed Mueller, calling it an "old and sick trick" to deter coalition strikes.
"The plane from the crusader coalition bombed a position outside the city of Raqa after Friday prayers," the Islamic State group said.
"No fighter was wounded but we can confirm that an American hostage was killed," it said on jihadist websites without publishing any pictures of her body.
Jordan is part of the international coalition battling the Sunni extremist group, which has seized swathes of territory in Syria and Iraq and imposed an extreme interpretation of Islam on the areas under its control.
Interior Minister Hussein Majali said in remarks published Saturday that the burning alive of a Jordanian pilot by IS marked a "turning point" in the kingdom's fight against extremism.
US authorities have never given figures on the number of Americans kidnapped in Syria, sticking to a policy of complete silence.
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