The group abducted the Coptic Christians after going room to room in the city of Sirte in the early hours of 3 January.
Reports from witnesses said the men separated the Muslims and let them go free before handcuffing and driving away with the Christians.
An article for Dabiq, the official Islamic State magazine, claims the 21 men would be executed in an act of 'revenge' for alleged violence towards women by the Coptic Church.
Pictures of the men wearing orange clothes and surrounded by masked gun men have also been published.
Bellow the image is the text "the humiliated followers of the Coptic Church".
Egypt has not commented on the pictures but it's reported the country is evacuating its citizens from Libya.
