Mohammad Rahim Hasanyar, a member of the provincial council, said the Afghan forces were in defence mode.
The main highway through Ghazni, the Kabul-Kandahar highway - has also been closed to traffic by the Taliban for days - which has affected hundreds of motorists.
The government in Kabul insisted that Ghazni was under its control but residents reported fierce clashes in several districts of the city and streets strewn with corpses.
On August 11, Defense Ministry spokesman Mohammad Radmanesh told RFE/RL's Radio Free Afghanistan that the city was under control of the security forces, but said clearance operations were continuing.
Local lawmaker Chaman Shah Ehtemadi told Reuters news agency: "Only the governor's office, police headquarters and intelligence agency's compound are in the hands of the government and Taliban are pushing to take them". He confirmed Taliban attacks on "multiple government centers" and noted the involvement of US aircraft in support of Afghan troops, who he said were holding their ground.
The insurgents began the attack by entering homes in Ghazni and then slipping out into the night to attack security forces.
In another incident, the Taliban attacked an Afghan military base in the central province of Daikundi with an explosive-laden vehicle, killing a number of government forces and taking at least 25 forces captive.
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The attack on Ghazni was the latest in a series of attempts by the Taliban to capture urban centers. The Afghan government claims to have killed dozens of Taliban fighters.
But Daniel Markey, a professor at the Johns Hopkins School of Advanced International Studies, said US policymakers should be concerned.
Earlier, residents described heavy gunfire ringing out and a government building set on fire.
"Overall, the situation is under our control in Ghazni, and the problems are not that serious", Najib Danish, spokesman for the Ministry of Interior, said at a news conference. At the height of the troop surge ordered by former U.S. President Barack Obama, thousands of North Atlantic Treaty Organisation troops were deployed in the city and province, including more than 2,500 from Poland. The Taliban pushed deep into the strategic city about 120 kilometres (75 miles) from the capital, Kabul.
As the USA and Taliban explore the next round of direct talks, attacks like the one Friday on Ghazni are a reminder of the pressure faced by Afghanistan's security forces.
According to the US military headquarters in Kabul, US aircraft conducted at least four air raids on Sunday and five more on Saturday, but details of the fighting were unclear as most of the city's telecoms masts were destroyed in recent fighting over recent.
Afghan President Ashraf Ghani is said to be considering a cease-fire offer for the upcoming Muslim holiday of Eid al-Adha.
"We stand ready to create all necessary conditions, at any stage of the peace process, to arrange direct talks between the government of Afghanistan and Taliban movement", Mirziyoyev told an worldwide peace conference in the Uzbek capital in March.