Shell explosions and sporadic bursts of machinegun fire were heard at the camp on Friday.
The army has been fighting in the outskirts of the camp but has not entered its official boundaries.
Lebanese security forces are not allowed to enter the 12 Palestinian camps in Lebanon under a 1969 Arab agreement.
"The Palestinian refugees are not being treated properly by Lebanon"
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The Lebanese army wants the fighters to surrender unconditionally and face justice for attacking and killing its troops.
Fatah al-Islam says it is fighting in self-defence and has vowed to fight to the death.
Mainstream Palestinian factions have proposed deploying a force of some 150 fighters to secure part of the camp not held by Fatah al-Islam and create a "safe haven" for refugees still there, Palestinian political sources said.
They said they were still waiting for the army's response to the idea.
Most of the camp's 40,000 residents fled to the nearby Beddawi camp in the early days of the fighting, which erupted on May 20.
King Abdullah of Saudi Arabia donated $10 million to be distributed to Palestinian families hit by the conflict, Fuad Siniora, the Lebanese prime minister, said.