Three soldiers and two Palestinian civilians were wounded at the camp, medics said.
The army brought in reinforcements and deployed armoured vehicles mounted with heavy machine guns, and calm was restored for a few hours.
But fighting flared again as night fell on Sunday, Hanna reported.
Nahr al-Bared clashes
On Sunday, Lebanese military helicopters fired rockets and machinegun barrages at targets on the Nahr al-Bared refugee camp's coastal side.
The government of Fouad Siniora, the prime minister, has said it aims to eliminate Fatah al-Islam once and for all.
Al Jazeera's Zeina Khodr, reporting from Tripoli, said Fatah al-Islam confirmed that Abu Riyadh, one of its leaders, was killed by an army sniper on Saturday.
She said there had been conflicting reports in previous days that other senior leaders had been killed, reports that were denied by the group.
She the Lebanese army had managed to reach a few hundred metres inside the refugee camp and clear out pockets of fighters.
Machinegun fire could be heard outside the camp on Sunday as the army continued its siege backed by helicopter gunships and tanks.
Security sources said that two Lebanese army soldiers had been killed overnight, bringing the number of troops killed since Friday to eight.
More than 16 people, fighters and civilians, have died in the camp over the weekend.
Fatah al-Islam said it had lost three fighters.
Lebanon's official news agency said four fighters were killed in Nahr al-Bared on Sunday, including the group's deputy leader named as Shehab Kaddur.
However, a man who Al Jazeera said was Kaddur denied he had been killed and boasted that his fighters "have arms that would shock the enemy".
Lebanon's An-Nahar daily reported that arrested members of Fatah al-Islam had confessed the group was planning a September 11-style attack.
"Fatah al-Islam planned to attack a large hotel in the capital using four suicide truck bombs at the same time as launching suicide attacks on embassies in east and west Beirut," the paper said, without giving its sources.
Abu Salim Taha, Fatah al-Islam spokesman, told Al Jazeera: "We are inflicting great damage on the part of the Lebanese army.
"We are ... in total control of the battlefield... We have the upper hand in fighting at the moment. We will never surrender... we will fight till the last drop of blood."