The statement identified the woman as Kayla Jean Mueller and said she was killed during Muslim prayers -- which usually take place around midday on Fridays -- in airstrikes that targeted "the same location for more than an hour."
No Islamic State militants were killed in the airstrikes, the statement further claimed.
It published photos allegedly of the bombed site, showing a severely damaged brown colored three-story building -- but no images of the woman.
The White House said it could not confirm the reports. "We are obviously deeply concerned by these reports. We have not at this time seen any evidence that corroborates ISIL's claim," said Bernadette Meehan, spokeswoman for the White House National Security Council, in a statement.
State Department spokeswoman Marie Harf said: "I cannot confirm those reports in any way. Obviously people are looking into them, but cannot confirm them." When asked if she knew if the hostage was alive, Harf said: "We're just not going to get into specifics about Americans being held overseas."
The IS statement could not be independently verified. It appeared on a militant website commonly used by IS and was also distributed by IS-affiliated Twitter users.
Jordan said it was highly skeptical about claims by Islamic State that an American woman held hostage by the militants in Syria had been killed in a bombing raid by Jordanian fighter jets.
"We are looking into it but our first reacion is that we think it is illogical and we are highly skeptical about it... It's part of their criminal propaganda," government spokesman Mohammad Momani said.
"How could they identify Jordanian war planes from a huge distance in the sky? What would an American woman be doing in a weapons warehouse?" he added.
Mueller, of Prescott, Arizona, had been working in Turkey assisting Syrian refugees, according to a 2013 article in The Daily Courier, her hometown newspaper. The 26-year-old 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 local paper, Mueller had been working with the humanitarian aid agency Support to Life, as well as a local NGO that helped female Syrian refugees develop skills.
A 2007 article about Mueller from the same local newspaper said she was a student at Northern Arizona University and was active in the Save Darfur Coalition.
On Sunday, President Barack Obama said the US was "deploying all the assets that we can" to find Mueller.
"We are in very close contact with the family trying to keep them updated," he said in an interview with NBC's Today Show. "Obviously this is something that is heart-breaking for the family and we want to make sure we do anything we can to make sure that any American citizen is rescued from this situation." Her identity had not been disclosed out of fears for her safety.
If her death is confirmed, she would be the fourth American to die while in the captivity of the Islamic State militants. Three other Americans, journalists James Foley and Steven Sotloff and aid worker Peter Kassig were beheaded by the group.
Jordan, which is part of a US-led coalition bombing Islamic State group targets in Syria, stepped up its attacks after IS announced it had killed a captive Jordanian pilot. 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.
There was no word from the Jordanian government on whether its planes had struck Raqqa on Friday. But activists who monitor the Syrian conflict from inside the country said US-led coalition planes hit several targets on the edges and outskirts of Raqqa, in quick succession on Friday.