Yarmouk, set up as a camp for Palestinian refugees decades ago and now a residential area housing Palestinians and Syrians, has been besieged by government forces for nearly two years.
It has been the scene of fierce clashes between groups taking rival sides in the Syrian conflict.
Shortages of food, water and medical supplies have caused serious suffering in the area where an estimated 18,000 civilians remain.
Conditions have worsened since ISIL entered the camp last Wednesday, attacking a Palestinian group opposed to the Syrian government. Government forces have since bombarded the area, even using barrel bombs, activists say.
Pierre Krahenbuhl, commissioner-general of UNRWA, the UN agency aiding Palestinian refugees, described the situation in Yarmouk as "one of the most severe ever".
The Jafra Foundation, a civil relief organisation active in Yarmouk, told Al Jazeera that 13 civilians have been killed in the last week.
The group said most civilians remained trapped inside their homes, without access to food and water, as ISIL snipers were stationed on rooftops.
However, about 500 families were able to leave Yarmouk over the weekend, entering Yelda - a neighbouring area also besieged by government forces but without ISIL presence.
As of February 2015, OCHA officially recognised 11 besieged areas in Syria with the estimate of 212,000 civilians living in them. A Syrian-American NGO, the Syrian Medical Society, estimates that the number of people living under siege is as high as 640,200.
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