St Elijah's in Mosul was more than 1,400 years old.
Iraqi monk, Father Dankha Issa (pictured far right with refugees), told the Catholic charity, Aid to the Church in Need, the monastery was: "A symbol of the Christian presence in Iraq.
"The fact that it has been destroyed is terrible."
Analysis of satellite pictures suggests the monastery had been destroyed between August and September 2014, just two months after the city was seized by jihadis.
Before then, thousands of Christians had been living in the predominantly Sunni city in northern Iraq.
Father Dankha said the monastery: "Had stood abandoned for a long time - but it meant a lot to us Christians.
"It was an expression of our extremely long history in Iraq.
"It is the sons of the devil who do such work. We can only pray for them. God alone can help us."
The Islamic State group has destroyed a large number of sites of religious and cultural significance in Iraq and Syria.