The video appeared on the Twitter feed of a website that supports Islamic State.
After the Islamic State released photos of the kidnapping Coptic Christians last week, Egypt offered to evacuate thousands of its citizens from Libya.
NPR's Leila Fadel reports for our Newscast unit that ISIS claims to have kidnapped the 21 men last month. "The brutal extremist group appears to be setting up shop in the security vacuum that is Libya today," she says. "In the video, a masked man says in English that the group plans to conquer Rome."
Reuters, quoting Egypt's state television, says President Abdel Fattah al-Sisi has called a seven-day mourning period and called an urgent meeting of the country's top military commanders.
"Egypt said it had no choice but take urgent measures on Friday with Isis expanding fast across Libya, its war-stricken neighbour. 'We are helping them,' Cairo's foreign ministry spokesman, Badr Abdelatty, said. 'If they would like to come back from Libya, we are facilitating their return to their home country.'
"The Coptic Christian guest workers were kidnapped in Sirte, on Libya's coast, by Isis units based there in two operations in December and January."