The London-based Syrian Observatory for Human Rights says ISIS and the al Qaeda-affiliated Al-Nusra Front took control of 90% of the camp in southern Damascus.
Calling the lives of Yarmouk refugees "profoundly threatened" on Sunday, the United Nations Relief and Works Agency issued a statement urging humanitarian aid access. "Never has the hour been more desperate in the Palestine refugee camp of Yarmouk," the statement said.
The UNRWA estimates 18,000 civilians remain trapped in the camp that has been engulfed in fighting between the government and rebel forces since December 2012.
Syria's state-run SANA news agency reports up to 2,000 people have fled in the past two days as food, water and medical supplies remain scarce.
"All people are trying to leave the camp," says Syrian activist Abu Mohammed in Damascus who used to live in Yarmouk.
"There is no electricity," says Mohammed. "ISIS controls the hospital so injured people have nowhere to go."
The Syrian Observatory for Human Rights reports barrel bombs were dropped on the camp Sunday as clashes continued.
The Palestine Liberation Organization called on international bodies to assist in the evacuation of people from the camp. "Reports of kidnappings, beheadings and mass killings are coming out from Al- Yarmouk, which is under a brutal campaign of murder and occupation," Palestine Liberation Organization Executive Committee Member Dr. Saeb Erekat said Saturday.
Yarmouk, the largest Palestinian refugee camp in Syria, was formed in 1957 to accommodate people fleeing the Arab-Israeli conflict.
"The levels of humanity that we have seen have now descended into further levels of inhumanity," said Chris Gunness, spokesman for the UNRWA. Yarmouk, he added, "was always a place where human rights meant very little. We are seeing it descend further."