In Jordan, meanwhile, thousands of people marched to demand retribution against IS for murdering a captive pilot.
The jihadists' claim came as Amman said dozens of its jet fighters had struck IS, widening their campaign from Syria to include targets in neighbouring Iraq.
Britain-based monitor the Syrian Observatory for Human Rights said more than 30 IS jihadists were killed in coalition raids yesterday around Raqa, the "capital" in Syria of the jihadists' self-proclaimed caliphate.
"The plane from the crusader coalition bombed a position outside the city of Raqa after yesterday prayers," it said.
"No fighter was wounded but we can confirm that an American hostage was killed in the strikes.
"The family of aid worker Mueller, 26, urged media restraint in their reporting of the IS claim.
Mueller was captured in August 2013 in Aleppo, while working for a Spanish Medecins Sans Frontieres (Doctors Without Borders) hospital, the family statement said.
"An old and sick trick used by terrorists and despots for decades: claiming that hostages human shields held captive are killed by air raids," Judeh tweeted.
The IS statement did not show any pictures of a body and there was no independent confirmation of the claim.
The United States said it has not yet seen any proof to confirm the IS claim.
"We are obviously deeply concerned by these reports. We have not at this time seen any evidence that corroborates ISIL's claim," said National Security Council spokeswoman Bernadette Meehan, using another acronym for IS.
Amman's government spokesman Mohammad al-Momani dismissed the jihadists' claim as "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," Momani told AFP.
Earlier, thousands of people marched in Amman in solidarity with Maaz al-Kassasbeh, the young Jordanian airman whose burning alive IS showed in a video released this week.
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