The UN Security Council has demanded humanitarian access to the Palestinian refugee camp of Yarmouk in Damascus.
One UN official described the situation for the 18,000 refugees there as “beyond inhumane”.
The situation has deteriorated since 1 April, when Islamic State launched an offensive.
Palestinian militiamen opposed to the Syrian government and some Free Syrian Army fighters are leading the fight against the IS militants.
‘This has to stop’
The chair of the 15-member Security Council, Jordan’s ambassador Dina Kawar, called for the “protection of civilians… humanitarian access… and life-saving assistance”.
Delivering a report to the council, Pierre Krahenbuhl, of the Palestinian Unwra relief agency, said the situation was “more desperate than ever”.
The Palestinian UN ambassador, Riyad Mansour, said that saving the refugees was his government’s main priority.
He appealed to member nations to relocate the refugees elsewhere in Syria or abroad.
Analysis: Jim Muir in Beirut
Because of its strategic location, the Syrian government has always feared Yarmouk might be used as a springboard for a rebel thrust into the heart of the capital.
With some reports saying IS may now control as much as 90% of the area, that brings the militants closer than ever to the centre of Damascus.
Government helicopters are reported to have been dropping barrel bombs on Yarmouk, although rebel groups accuse the regime of encouraging the spread of Islamic State, so it can dismiss the entire rebel movement as a bunch of terrorists.
Ahmed Majdalani, an official with the Palestine Liberation Organisation, said a delegation was on its way to Damascus to discuss a humanitarian corridor with the Syrian government and Palestinian factions in the camp.
Another Unwra official, Chris Gunness, said: “The situation in the camp is beyond inhumane.
“There is no food, there is no water and there is very little medicine… People are holed up in their houses, there is fighting going on in the streets. There are reports of… bombardments. This has to stop and civilians must be evacuated.”
Sunday’s heavy fighting turned into sporadic clashes on Monday, monitors said.
Monitors say IS and the al-Qaeda-affiliated Nusra Front, who have fought each other in other parts of Syria, were working together in Yarmouk.
A few hundred people were reported to have managed to escape the camp over the weekend.
Syria’s bloody conflict, which has entered its fifth year, has claimed the lives of more than 200,000 Syrians.
The battle between forces loyal to President Bashar al-Assad, rebels opposed to his rule and Islamic State has also driven more than 11 million people from their homes.
Yarmouk was first built for Palestinians fleeing the 1948 Arab-Israeli war.
Before the Syrian civil war began, it had more than 150,000 refugees living there and had its own mosques, schools and public buildings. But since 2012, the camp has been besieged by fighting.