It had initially been thought 20 Egyptian believers were taken in the city of Sirte in the early hours of Saturday morning but state media has revised the number down.
Armed gunmen stormed a housing complex and separated the Muslims and let them go free before handcuffing and driving away with the Christians.
Fr. Philemon Attia, a priest of the Mar Girgis church in Beni Suef, said his nephew was among the men taken.
"One of our relatives who is still in Libya and lives in the same housing building called us today and told us about the abduction my nephew and twelve others," he told International Christian Concern.
"At dawn some armed, masked men stormed the housing building and broke the doors of their rooms and abducted them at gunpoint."
Fr. Philemon said he had been told a group with links to Islamic State may be behind the kidnappings.
IS has been persecuting tens of thousands of Christians as it takes over large parts of Iraq and Syria trying to create a caliphate that is ruled by strict Muslim law.
The priest said many of the Christians at the housing complex "stayed in their housing building for more than 25 days like captives and they couldn't go out of their rooms because they heard that the area where they were living was controlled by the Islamic State group.
"They believed their lives would be in danger if they went out. Every time I called him he was asking me to pray for them," he added.
The kidnappings come just five days after seven Coptic Christians were abducted at a check point as they tried to return to their home country of Egypt.