An officer at the camp said the fighters had put up less resistance over the past two days and now controlled an area of about 22,500 square metres.
The officer also said the army was setting fire to buildings around the area still controlled by the group to smoke them out.
The army was also clearing the narrow and winding streets of mines and booby traps that have made the troop advance difficult, he said.
"Our soldiers are clearing buildings seized in the past three days from the last sector still controlled by the Islamists," the spokesman said.
He said that although the soldiers were advancing slowly, they would "very soon claim victory over the Islamists".
The number of Lebanese soldiers killed in the fighting is now 121.
There is widespread speculation that the army may be pushing to end the standoff by August 1, to coincide with the country's Army Day celebrations, but the spokesman said no deadline has been set.
On Wednesday, the army stepped up its shelling of Fatah al-Islam positions in a push to end the battle that began on May 20 and has cost the lives of more than 200 people.
|Much of Nahr al-Bared has been levelled [AFP]|
Almost all of the camp's 31,000 mostly Palestinian residents have been evacuated
However, about 20 wives and 45 children of the fighters have stayed inside the besieged settlement despite appeals by the military for them to leave the largely destroyed camp.
On Friday, about a dozen rights activists, some of them Westerners, demonstrated near the entrance to the camp, asking the army to spare the lives of the civilians still inside.
The showdown between the two sides began after the al-Qaeda-inspired fighters launched a series of attacks against the army.
The fighting since has exacerbated political tensions in Lebanon where the government has been paralysed by an eight-month standoff between the country's feuding parties.