The British government says there is "strong reason to believe" that a British citizen has been executed by al Qaeda militants in North Africa.
Edwin Dyer was kidnapped in Niger in January, but was being held in Mali, the BBC reports.
In a statement posted on an Islamist website, the group said it had killed Mr Dyer on 31 May after the British government failed to respond to its demands.
The group had sought the release of Abu Qatada, a Islamist from Jordan, who has been imprisoned in Britain since 2005.
Abu Qatada, named by a Spanish judge as the right-hand man in Europe of Osama bin Laden's al Qaeda network, denies belonging to the group.
Abu Qatada is awaiting extradition to Jordan, where he was convicted of terrorism offences in his absence and faces life in jail. In February, Britain's highest court ruled that he could be deported to Jordan, despite fears he may be tortured.
British Prime Minister Gordon Brown on Wednesday condemned what he called an "appalling and barbaric act of terrorism".
Al Qaeda in the Islamic Maghreb (AQIM) has claimed responsibility for the kidnapping of two Canadian diplomats and four European tourists in the past five months. The two diplomats and two of the tourists were freed in Mali in April.
In May, Algerian media reported AQIM was demanding 10 million euros in exchange for Mr Dyer and another hostage, a Swiss national, being held in the Sahara.