"The fighting has begun. We are attacking the last Daesh positions in district three" where the jihadists are cornered, a GNA fighter told AFP, using an Arabic acronym for IS.
The GNA forces media centre said the new push had begun to retake Sirte, located 450 kilometres (280 miles) east of the capital Tripoli.
"Our forces are advancing inside the areas where Daesh is, in district three, and so far have taken control of" two banks and a hotel, the media centre said on its Facebook page.
It also said they had thwarted an attempted suicide bombing.
Ten members of the GNA forces were killed and 60 wounded in Saturday's clashes, said a doctor at hospital in Misrata, a city half-way between the Sirte and Tripoli where the casualties are taken.
The loyalists used tanks and heavy artillery to dislodge snipers posted in apartment blocks before troops advanced on foot, said an AFP journalist in the city.
Ambulances earlier streamed out of Sirte -- hometown of dead dictator Moamer Kadhafi -- headed for Misrata.
The fighting eased after sunset, the journalist said, with sporadic gunfire on the ground and military aircraft heard overhead.
Since the offensive against Sirte began on May 12, more than 400 fighters loyal to the government have been killed and about 2,500 wounded.
It is not yet known how many IS militants have been killed, but the GNA media centre said the bodies of 10 jihadists had been found in a school in district one, which was being combed after being retaken on Monday.
The forces loyal to the UN-backed GNA had said last weekend they were preparing to "liberate" the entire city after seizing several IS positions, including its headquarters.
On Wednesday, GNA head Fayez al-Sarraj visited Sirte for the first time since loyalist forces launched their offensive more than three months ago.
- US air raids -
Sarraj and some of his ministers toured former front lines as well as the Ouagadougou conference centre which IS had used as its base.
"We will continue to chase, with the help of God, the Daesh remnants and strike them wherever they may be in our country," Sarraj said this week.
The capture of Sirte by IS last year sparked fears the jihadists would use it as a springboard for attacks on Europe.
The Sunni extremists took advantage of the chaos in oil-rich Libya after the 2011 uprising to seize Sirte in June 2015, hoisting their black flag above the city.
The offensive on the ground has been backed by US air power.
On Friday, the United States Africa Command said that since the US campaign began on August 1, US drones, helicopters and bombers had carried out a total of 108 air strikes against the jihadists in Sirte.
It said that on August 31, targets including five "enemy fighting positions" and a vehicle bomb were hit.
Fewer than 200 IS jihadists remain in Sirte, Pentagon spokesman Captain Jeff Davis said on Thursday, and they are essentially surrounded by GNA forces and the sea.
The GNA has been struggling to assert its control over all of Libya.
France on Friday urged Sarraj to find a compromise with the Tobruk-based parliament in the far east of the country, which does not recognise the unity government.
"He must find a compromise with the Tobruk parliament and General (Khalifa) Haftar," who controls the armed forces in the east, Foreign Minister Jean-Marc Ayrault said in Paris.