The militants said it was in revenge for recent US air strikes against the Islamic State group in Iraq.
The video has not been independently verified, but the White House said if it was genuine, the US would be "appalled by the brutal murder".
Foley's family wrote on Facebook (external - login to view): "We know that many of you are looking for confirmation or answers. Please be patient until we all have more information, and keep the Foleys in your thoughts and prayers."
If (the video is) genuine, we are appalled by the brutal murder of an innocent American journalist”
End Quote Caitlin Hayden US security spokeswoman
Foley has reported extensively across the Middle East, working for America's Global Post and other media outlets including the French news agency AFP.
In the video titled A Message to America, a man who appears to be James Foley is dressed in orange kneeling in desert-like terrain beside an armed man dressed in black.
The militant, who speaks with a British accent and identifies himself as an IS member, says the journalist's death is the direct result of US bombing of IS targets in Iraq.
The US has recently carried out a number of air strikes against IS militants, who have swept through large swathes of north-western Iraq.
In a statement, the Global Post ask for "prayers for Jim and his family", adding that it was waiting for the video to be verified.
US officials confirmed that they had seen the video.
White House National Security Council spokeswoman Caitlin Hayden said: "If genuine, we are appalled by the brutal murder of an innocent American journalist and we express our deepest condolences to his family and friends."
The footage claimed to have been released by the IS also shows another prisoner identified by the militants as an American reporter.
IS supporters have recently threatened to attack American citizens, saying: "We will drown all of you in blood."
The militant group has been accused of massacring hundreds of people in areas under their control in Iraq and also in eastern Syria.
The violence has displaced an estimated 1.2 million people in Iraq.
Pursuing an extreme form of Sunni Islam, IS has persecuted non-Muslims such as Yazidis and Christians, as well as Shia Muslims, whom it regards as heretics.
BBC News - Islamist militants 'kill reporter James Foley on video' (external - login to view)