There are some 88,000 registered Iraqi refugees in Syria, most in Damascus, along with about 8,000 refugees from other nations such as Somalia and Afghanistan, the agency said.
Thousands of the refugees, mainly Iraqis who have been living in the Damascus suburb of Seida Zeinab, have fled their homes due to the increasing violence and "targeted threats" against them, refugee agency spokeswoman Melissa Fleming said.
One family of seven, including children, was found "murdered" in their Damascus apartment, shot dead at close range by unknown assailants, said Fleming, while another three Iraqi refugees also were killed by gunfire last week.
The Iraqi government had sent eight flights to Damascus and evacuated at least 750 residents by Friday morning, and it planned to send more, according to Iraqi and U.N. officials. On Tuesday, Iraq's government called for all its citizens living in Syria to return home immediately to escape the civil war.
Between 2005 and 2007, thousands of Iraqis fled to Syria to escape widespread sectarian fighting during the worst of violence in their homeland. The U.N.
And on Monday, the bodies of two Iraqi journalists killed in Syria were handed over to Iraqi authorities. The two journalists died in the Damascus suburb of Jaramana and there had been no fighting there at the time.