Afghan forces said they remained in control of the strategically-important city, but the militants reportedly came within a few hundred yards of several government buildings.
The city was in lockdown and fighting continued throughout the day Friday, with sporadic bursts of gunfire from Taliban fighters who had hunkered down in elevated positions inside Ghazni from which they were shooting, some residents said, speaking on condition of anonymity for fear for their safety.
He said the Taliban militants have sustained heavy casualties during the attack on Ghazni city and in clashes with the security forces but added that no specific information is available regarding the exact toll.
Defence Ministry spokesman Mohammad Radmanish later said the army supported police and the city was now under control of government forces.
At least 39 insurgents were killed, while 14 police died and 20 were wounded in the fighting, said provincial Police Chief Farid Ahmad Mashal.
According to the officials, at least 26 security personnel are among those wounded during the clashes.
Taliban spokesman Zabihullah Mujahid claimed parts of the city had been seized and scores of people killed.
"US forces responded with close-air support (US attack helicopters) and conducted one strike (drone)". Afghan officials said only one of its fighters was confirmed killed, but a Taliban spokesman said 140 enemy forces were dead.
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As fighting continued throughout the day, he said USA forces returned, sending attack helicopters and fighter jets back to the area.
The Taliban had called in fighters from central Maidan Wardak and southern Zabul province and they attacked security posts, the police headquarters and other military installations.
Mohammad Rahim Hassanyar, a senator from Ghazni, said the operation to expel all Taliban fighters was "very slow".
After dawn Friday, Ghazni's residents were staying indoors and all shops in the city remained closed.
The Taliban have stepped up attacks across the country since North Atlantic Treaty Organisation and the United States formally ended their combat mission in 2014, and have seized control of several districts.
The assault comes as pressure builds on the insurgents to enter peace talks.
It has so far ignored President Ashraf Ghani's offer of unconditional peace negotiations.
Pompeo also said the role of foreign forces in Afghanistan would be on the table.
Anticipation has also been mounting about the possibility of a government ceasefire announcement for the Islamic holiday of Eid-al Adha later this month. Washington has repeatedly refused, saying negotiations must be Afghan-led. Islamabad had acted as a broker for the Taliban, helping to set up negotiations with both China and the United States, the Financial Times reported.