A video posted online claims to show that Islamic State militants have killed the captured US aid worker Abdul-Rahman Kassig.
The video shows a masked man standing over a severed head which he says is that of Mr Kassig.
The US says it is working to determine the authenticity of the video, which also shows a mass beheading of 18 Syrian troops.
Mr Kassig, also known as Peter, was captured last year.
His family, who live in the US state of Indiana, said they were awaiting confirmation of the reports about their "treasured son".
"We prefer our son is written about and remembered for his important work and the love he shared with friends and family, not in the manner the hostage takers would use to manipulate Americans and further their cause," the family said in a statement.
Analysis: Frank Gardner, BBC security correspondent
This is perhaps the most elaborate and graphic of all the murder videos posted online by IS. Much of it is taken up with a recent history lesson on Iraq and Syria as seen through the eyes of the jihadists. But the latter part shows the beheading of 18 Syrian prisoners in revolting, lingering detail.
Unlike earlier videos, this one revels in gore. Amongst the boiler-suited captives murdered in cold blood is a man IS says is the former US soldier Peter Kassig, who converted to Islam and changed his name to Abdul-Rahman.
Neither his conversion, nor the fact that he was helping refugees when he was captured a year ago, appear to have saved him.
Mr Kassig's murder is a sign of frustration that IS militants are unable to hit back at the coalition air strikes that have driven them off key sites like the Mosul and Haditha dams, and prevented them from seizing the town of Kobane.
Unlike similar videos released in the past, the latest shows the faces of many of the jihadis. It also at one point gives a specific location - near Dabiq in Aleppo province, with an identifiable village in the background.
The latest video did not show the person identified as Mr Kassig being beheaded. Also unlike previous videos, it did not show other Western captives or directly threaten to behead anyone else.
The US National Security Council said it was investigating the video, and would be "appalled by the brutal murder" if it was confirmed.
IS has previously murdered four Western hostages - British men Alan Henning and David Haines, and US journalists James Foley and Steven Sotloff.
Those killings were carried out by a man believed to be British. That man bears a resemblance to one of the masked militants pictured in the video purporting to show Mr Kassig.
British Prime Minister David Cameron said he was "horrified by the cold-blooded murder" and that Islamic State had "again shown their depravity".
Mr Kassig's parents last month released extracts of a letter written by their son, in which he said the "stress and fear" of captivity were "incredible".
"They tell us you have abandoned us and/or don't care but of course we know you are doing everything you can and more.
"Don't worry Dad, if I do go down, I won't go thinking anything but what I know to be true. That you and mom love me more than the moon and the stars."
Mr Kassig was a former US Army Ranger who served in Iraq.
He later trained as an emergency medical technician and founded the Special Emergency Response and Assistance (Sera) organisation, helping to supply camps on both sides of the Syrian border.
He was undertaking a project for Sera when he was captured in October 2013 while travelling to eastern Syria.
The exiled leader of Syria's opposition National Coalition, Hadi al-Bahra, said in a statement Mr Kassig would be remembered as a hero.
"Kassig paid the ultimate sacrifice trying to relieve the suffering of his fellow human beings far away from home. The thugs of Isis will pay the price for their evil crimes," he said.