The United Nations Relief and Works Agency said at least half of Yarmuk's population of more than 112,000 had fled the violence rocking the 2.1 square-kilometre (about one square mile) camp.
An AFP correspondent at the border reported an influx of hundreds of Palestinians into neighbouring Lebanon in the face of the fierce fighting between Syrian rebels and their Palestinian allies, and Palestinian factions still loyal to Damascus.
President Bashar al-Assad's forces carried out air raids on the eastern and southern outskirts of the capital, killing three civilians, and clashes also erupted near the airport road, said the Syrian Observatory for Human Rights.
Fighting had rocked the Yarmuk camp during the night as the insurgents backed by some Palestinian fighters launched an offensive to push out a pro-regime Palestinian faction.
One resident reported seeing hundreds of fighters of the rebel Free Syrian Army from neighbouring districts inside the camp, while the army was nowhere in sight.
The mosques of the camp broadcast an army ultimatum giving the 150,000 residents until 1000 GMT to leave their homes, he told AFP.
"Some have decided to comply, but others chose to stay," the resident said.
Al-Watan newspaper reported the army was gearing up for a major assault.
"Soldiers have massed in large numbers and are preparing a military operation to cleanse the camp," the pro-regime daily said.
An AFP journalist at the scene said army checkpoints were set up at the northern and southern entrances.
Soldiers were preventing anyone from entering the camp, while dozens of families could be seen with suitcases in hand waiting to leave aboard pickup trucks.
The population of the camp had halved by Tuesday, according to UNRWA.
"Our conservative estimate would be that at least 50 percent of the people living in the Yarmuk suburb have left or are leaving, taking refuge in other parts of Damascus, UNRWA schools and facilities or leaving altogether," spokesman Sami Mshasha said.
Many have fled to neighbouring Lebanon.
A large number of buses and cars filled with Palestinians were waiting to enter Lebanon through the Masnaa border crossing on Tuesday, an AFP photographer said.
Palestinians in Syria, who are mainly Sunni Muslims, are divided over the uprising against the rule of President Assad, who is from the Alawite offshoot of Shiite Islam but whose regime has long given shelter to hardline militant factions.
Abu al-Sakan said a group of Palestinian militants had defected to the rebel side.
Syrian Observatory director Rami Abdel Rahman said: "There are definitely Palestinians fighting with the rebels."
Ekmededdin Ihsanoglu, head of the Organisation of Islamic Organisation, said "neither the regime nor the opposition can be excused" for the clashes at the camp.
The World Food Programme warned that the spiralling violence across Syria was making it increasingly difficult to distribute food to trapped civilians.
"Food needs are growing in Syria," said WFP spokeswoman Elisabeth Byrs, whose organisation distributes most of its aid through the Syrian Arab Red Crescent (SARC).
Citing SARC figures, Byrs told reporters in Geneva that nearly 2.5 million people currently need emergency food assistance yet the WFP is only able to reach 1.3 million people each month.
Turkish Foreign Minister Ahmet Davutoglu urged Syria's key regional ally Iran to "send a clear message" to the Assad regime to stop the violence against its own people.
One of the highest-profile US journalists to report from Syria, Richard Engel of NBC, was meanwhile freed after a firefight between his pro-regime militia captors and rebels fighters.
NBC said Engel, 39, and his TV crew went missing shortly after crossing into Syria from Turkey on Thursday, and that it had not been able to contact them until it learned they had been freed on Monday.