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Engineer Mr Bigley, 62, and two Americans with whom he shared a house in the wealthy al-Mansour district of Baghdad, were captured on 16 September by the Islamist Tawhid and Jihad group.
His fellow contractors Eugene "Jack" Armstrong and Jack Hensley were beheaded on 20 and 21 September, when their kidnappers' demands for the release of Iraqi women prisoners were not met.
The United States is holding two Iraqi female weapons scientists, Rihab Rashid Taha and Huda Salih Mahdi Ammash, but says it has no plans to release them.
Efforts to secure Mr Bigley's release have been stepped up in the past few days.
Libya's Colonel Gaddafi and Sinn Fein's Gerry Adams have appealed to his kidnappers on behalf of the family.
The Irish Government has also confirmed Mr Bigley can claim Irish citizenship. The Republic of Ireland opposed the war in Iraq and is therefore seen as neutral in the conflict.
British Foreign Secretary Jack Straw, who arrived in Iraq earlier in the week, is believed to have exchanged messages with the captors.
But the government insists it will not negotiate with terrorists or give in to their demands.
Several videos have been released showing Mr Bigley pleading for mercy and appealing to Prime Minister Tony Blair to agree to the kidnappers' demands.
Reports from Iraq claim that US-led forces were closing in on the kidnappers' location.
Tawhid and Jihad is led by Jordanian militant Abu Musab al-Zarqawi, who is believed to have links to al-Qaeda.
Another of the kidnappers is believed to be an Egyptian-born militant, who fought in Iraq against US troops during the invasion and then joined al-Zarqawi.
Mr Bigley was preparing for retirement in Bangkok with his Thai wife Sombat.
His 86-year-old mother, Lil, was taken ill earlier in the family's campaign for his release after appealing on television for her son to be freed.
Mr Bigley's three brothers, Stan, Paul and Phillip and his surviving son, Craig, also campaigned ceaselessly for his release.
His younger son Paul fell into a coma after being knocked down by a lorry in 1986. Mr Bigley and his first wife, Margaret, had to take the devastating decision to turn off the life support machine.
Kenneth Bigley's death was confirmed the following day, when a video released by his captors showed him being decapitated.
There was widespread condemnation of the killing and the Queen sent a private message of condolence to Mr Bigley's mother in Liverpool.
A brief statement delivered by Phillip Bigley said the government had "done everything it possibly could to secure the release of Ken in this impossible situation".
But another brother, Paul, accused Mr Blair of having "blood on his hands".
A memorial service was held in November 2004 which was addressed by the Prime Minister Tony Blair.
Mr Bigley's body has not been found.
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