The delegation arrived in Baghdad on Saturday to persuade Bigley's captors to spare his life.
Dr Daud Abd Allah and Dr Musharraf Husayn will urge Muslim leaders to put pressure on the captors.
Iqbal Sacranie, secretary-general of the Muslim Council of Britain, said: "Be merciful. Our religion Islam does not allow us to harm the innocent. We urge you to please listen to this appeal and release this man back into the arms of his waiting family."
Bigley, a 62-year-old civil engineer from Liverpool, was captured at gunpoint in Baghdad on 16 September with two American colleagues.
The two Americans have both since been beheaded by their captors.
There has been no word on Bigley or his captors since he was seen pleading for his life in a video posted on a website on Wednesday.
Bigley was last seen pleading for
his life on a website videotape
An appeal from Bigley's 86-year-old mother Lil, made in the family's home city of Liverpool, was broadcast on Aljazeera earlier this week.
The British Embassy in the Iraqi capital has also distributed 50,000 leaflets with pleas from Bigley's family for him to be set free.
The captors from the Tawhid and Jihad Group, led by al-Qaida linked Abu Musab al-Zarqawi, have threatened to murder him unless the UK and US release all women prisoners held in Iraqi jails.
Foreign Secretary Jack Straw has come under fire in his own constituency for his handling of the crisis as Britain continues to stand firm against negotiations with the captors.
"Be merciful. Our religion Islam does not allow us to harm the innocent. We urge you to please listen to this appeal and release this man back into the arms of his waiting family."
Muslim Council of Britain
Bigley's capture is a conundrum for British Prime Minister Tony Blair. He cannot be seen to be negotiating with the captors, but may face a backlash if Bigley is murdered and he is perceived to have not done enough.
In a statement Straw said: "We are very grateful to the Muslim Council of Britain and will do all we can to support their efforts. We all hope to see Ken safely back with his family''.
Members of the Muslim community in Blackburn, Lancashire, have organised a protest rally on Sunday to push for the government to do more for Bigley.
Meanwhile, more than 100 people attended a two-hour candle-lit vigil at Liverpool's Catholic Cathedral on Friday night in support of the engineer.