The UN agency for Palestinian refugees UNRWA issued an urgent plea for humanitarian access to the area, saying the situation in Yarmouk was a “source of universal shame.”
Militants from IS first attacked the camp, just seven kilometers (four miles) from central Damascus, on Wednesday.
They were initially repelled by Palestinian forces inside Yarmouk, but have since captured 90 percent of the sprawling area.
“The residents of the camp are between the devil and the deep blue sea,” said Ayman Abu Hashem, who heads the Syrian opposition interim government’s committee on Palestinian refugees.
“The camp is surrounded by the regime, and inside the forces of IS have almost completely taken over.”
Abu Hashem said Palestinian forces from the Aknaf Beit al-Maqdis group, which is loyal to the Palestinian movement Hamas, were effectively encircled by the militants.
And he said fears were running high that IS would take revenge in the camp because members of the group were expelled by residents and fighters last year.
The Syrian Observatory for Human Rights, a Britain-based monitor, said fundamentalist forces now controlled 90 percent of Yarmouk, which was once home to 160,000 people, both Syrian and Palestinian.
Observatory director Rami Abdel Rahman said Al-Nusra fighters were not battling alongside IS forces, but were also not opposing the group’s advances.
Palestinian officials accused Al-Nusra of facilitating IS’s entrance into Yarmouk, where the population has dwindled to just 18,000 residents.
Al-Nusra and IS are largely rivals, despite sharing a similar Wahhabi fundamentalist ideology, though there have been instances of local cooperation between fighters from the two movements.
Abdel Rahman said at least 18 people had been killed since the fighting in the camp began on Wednesday, six civilians and 12 Palestinian fighters.
Two fighters were beheaded, he said, and photos depicting two decapitated heads purportedly in the camp were circulated by fundamentalist militants on social media.
The Observatory also reported government air strikes on the camp, though there was no immediate information on casualties.
Yarmouk’s residents have suffered through regular clashes, government shelling and a regime blockade that has caused medical and food shortages.
Syrian forces remain outside the camp, and sources said troops had set up additional checkpoints around Yarmouk after the fighting began.
An agreement last June between the government and rebels, backed by Palestinian factions, led to easing of the blockade.
But humanitarian access has remained limited and was halted completely with the fundamentalist advance.
On Saturday, UNRWA spokesman Chris Gunness described the situation in the camp as “an affront to the humanity of all of us, a source of universal shame.”
“The humanitarian actors on the ground, like UNRWA, must have immediate access to bring urgent assistance to civilians,” Gunness said.
He urged “moral and diplomatic leadership from the international community” to ensure the protection of civilians.
Palestinian officials in the West Bank and Gaza also called for action, and the Syrian opposition National Coalition urged a safe passage for civilians.
The body called for “urgent intervention” from the Arab League and United Nations “to force the regime to open a safe passage for the safety of civilians.”
Ma’an staff contributed to this report.