At least one Afghan soldier has been killed and seven others wounded in the fighting in the southeastern city of Ghazni, provincial governor spokesman Arif Noori told AFP on Friday.
Civilian houses and army checkpoints have come under mortar attack and the bodies of dozens of Taliban fighters are in the streets, he added.
The Taliban began the attack late Thursday from several positions around the city, provincial police chief Farid Ahmad Mashal told AFP.
Residents who spoke to AFP said power has been cut to the area for hours since fighting erupted, with heavy gunfire ringing out across the city and a government building set on fire.
"We are scared for our life. The Taliban are roaming everywhere in and around the city," shopkeeper Mohammad Haleem told AFP.
Another resident, Yasan, said the Taliban were using loudspeakers at the mosque to warn residents to stay in their homes.
"Heavy explosions and gunfire can be heard. We are terrified," Yasan wrote in a Facebook post.
Police special forces have been deployed to help block the Taliban advance on the city, an Afghan security official said.
The Taliban issued a statement claiming to have captured "most of government buildings inside the city".
"So far 140 enemy forces have been killed or wounded," the group said.
The Taliban frequently exaggerate their battlefield gains and downplay losses incurred during fighting.
Growing pressure
It was the latest in a series of attempts by the Taliban over the past three years to capture urban centres.
Afghan forces have been struggling to hold back the resurgent militant group since the withdrawal of NATO combat forces at the end of 2014.
In May the Taliban attacked the western city of Farah. After a day of intense fighting, Afghan commandos and US air strikes drove the group to the outskirts of the city.
The attack on Ghazni comes as the Taliban faces growing pressure to agree to peace talks with the Afghan government to end the 17-year war.
It has so far ignored President Ashraf Ghani's offer of unconditional peace negotiations.
The Taliban has long insisted on direct talks with the United States. Washington has repeatedly refused, saying negotiations must be Afghan-led.
But there are tentative signs that diplomatic efforts to kick-start talks are starting to bear fruit.
Washington indicated a change in its longstanding policy in June when US Secretary of State Mike Pompeo said the United States was prepared to "support, facilitate and participate" in talks.
Pompeo also said the role of foreign forces in Afghanistan would be on the table.
Last month Taliban representatives met US officials for talks in Qatar.
Anticipation has also been mounting about the possibility of a government ceasefire announcement for the Islamic holiday of Eid-al Adha later this month.
An unprecedented truce in June brought fighting between security forces and the Taliban to a temporary halt, giving war-weary Afghans some welcome relief from violence.
