The sources said on Friday Bigley had been killed in the town of Latifiya, southwest of the Iraqi capital.
A video released by a purported Islamic group showed the beheading of the British captive, a Reuters witness said.
Bigley was shown making a statement as six of the captors stood behind him before one cut his head off with a knife.
The tape showed Bigley wearing an orange jump suit of the type worn by detainees in US prisons including the detention centre at Guantanamo Bay in Cuba.
Bigley was seized in Baghdad 22 days ago.
Bigley in the video made a
Earlier, Sky News television reported his death quoting sources in prime minister Tony Blair’s office, after Abu Dhabi television broke the news.
A Downing Street spokesman declined to comment on Friday when contacted by reporters, but the Foreign and Commomwealth Office(FCO) said it was investigating reports on Abu Dhabi television that Bigley was dead.
“We are aware of the report,” a foreign office spokesman told reporters in London, adding: “We have no further comment at this time.”
Bigley, 62, was seized in the Iraqi capital on 16 September along with two US colleagues who have since been beheaded by their captors, the armed Tawhid wa al-Jihad (Unity and Holy War) group.
Iraqi interior ministry officials in Baghdad said they had no information on the report.
Bigley’s brother Paul, who lives in Amsterdam, told Britain’s domestic Press Association news agency: “I have heard these reports and I am looking into them right now.”
“I have heard these reports and I am looking into them right now”
He added he wished to be left in peace while he investigated their authenticity.
Bigley’s family, from his ailing mother to his Thai wife, have issued emotional appeals for his release, to which Libyan leader Muammar al-Qadhafi added his voice on Wednesday.
Earlier in the week, Ireland – officially neutral during the Iraq war – announced that it had issued a passport to Bigley, who is part Irish, hoping that might sway his captors.
Tawhid wa al-Jihad initially called for the release of all women held in US-run prisons in Iraq, but more recently its demands have been unclear.