The extremist group posted photos on social media showing bulldozers destroying the Saint Eliane Monastery on Friday near the town of Qaryatain, which IS captured in early August.
A Christian clergyman in Damascus said that IS militants also wrecked a church inside the monastery that dates back to the first Christian centuries. The priest said the monastery included an Assyrian Catholic church.
"The church itself is built on the spot where [Saint Eliane's] father, who was a Roman officer some 1500 years ago, tried to force his son to renounce his Christian faith," said Christian Middle East commentator Dr Harry Hagopian on Premier's News Hour.
"His son refused so he was killed, and on the spot where he was killed was built this monastery some fifteen, sixteen hundred years ago which is known as Saint Eliane," he added.
Mr Hagopian also said he had heard reports that IS militants had opened up the tomb inside the monastery before they destroyed the building.
"They've also gone in there and taken out the body of the saint from the sarcoagus from the tomb in order to completely destroy the monastery and any relevance it has to the Christian faith." said Hagopian.
The Britain-based Syrian Observatory for Human Rights, which tracks Syria's conflict, also reported the destruction of the monastery.
Since capturing parts of Syria and Iraq, IS fighters have destroyed mosques, churches and archaeological sites.
On Tuesday the Islamic State group beheaded the 82-year-old retired chief archaeologist of Palmyra, after he refused to leave the ancient city.
Khaled al-Assaad was murdered execution-style on Tuesday afternoon in Palmyra, a UNESCO World Heritage site famed for well-preserved Greco-Roman ruins
Assaad was a leading expert on the ancient history of the city, which grew from a caravan oasis first mentioned in the second millennium BC
Palmyra was seized from government forces in May, fuelling fears the IS jihadists might destroy its priceless heritage as it had done in other parts of Syria and Iraq.
Listen to Middle East commentator Harry Hagopian speaking to Premier's Marcus Jones on the News Hour: