The status of their two young children, one a baby born in prison before Ishag's release, was not immediately known.
The couple were detained, for reasons that are unclear, at about 1100 GMT as they tried to leave the country, said the source.
He could not give more details except to say they were taken to a facility of the powerful National Intelligence and Security Service (NISS).
"She has the right to leave the country," the source said.
Her case sparked an outcry from Western governments and rights groups after a lower-court judge sentenced her to death on May 15.
Born to a Muslim father and an Ethiopian Orthodox Christian mother, Ishag was convicted under Islamic sharia law that has been in force in Sudan since 1983 and outlaws conversions on pain of death.
When Ishag was five, her father abandoned the family, and she was raised according to her mother's faith.
The Roman Catholic Archdiocese of Khartoum said she joined the Catholic church shortly before she married.