Mr Deya denies accusations that he ran a child trafficking ring on the pretext of praying for his followers to conceive via miraculous powers.
Reports say Mr Deya was arrested at the Edinburgh Sheriff Court, where he had gone to assist with a paternity case.
An official there reportedly recognised him as the wanted man. He is now in the custody of UK immigration officials.
Police in Kenya had issued a warrant of arrest for him after his wife, Mary, was arrested in Nairobi in September and charged with involvement in child trafficking.
She was arrested after she arrived at a hospital with a newborn baby to which she claimed to have given birth.
Doctors who examined Mrs Deya, 57, said her claim of recently having given birth was false.
Kenyan police allege the Gilbert Deya Ministries is an international baby-snatching ring and last year asked the UK government to extradite Mr Deya.
They say their investigation revolves around the disappearance of babies from Nairobi’s Pumwani Maternity Hospital and involves suspects in Britain, Ghana, Nigeria, Uganda and Kenya.
In 2004, the Kenyan authorities took 20 of Mr Deya’s “miracle babies” into care after they were found to have no genetic link to the women claiming to be their mothers.