"A total of 50 Taliban insurgents under Taliban key commander Mullah Toofan laid down arms and surrendered to local authorities in Qadis district today morning", Amini told Xinhua.
But people inside Ghazni say it has been overrun, with very little still under government control.
Helmand is among the relatively volatile provinces in South of Afghanistan where the anti-government armed militants are actively operating in some of its districts and often carry out terrorist related activities.
A spokesman for United States forces in Afghanistan also described the fight for Ghazni as a clearance operation, with sporadic clashes between security forces and insurgents punctuating relative calm.
However, communication lines with the city have been disconnected for about three days and little information has been gleaned in the time.
"We're winning", said Taliban spokesman Zabihullah Mujahid, adding that his group's fighters were seeking to rid Ghazni of government forces and include it in the 90 percent of the surrounding province he claimed militants already controlled... USA air support was also used to help Afghan troops retake Farah.
Mohammad Rahim Hasanyar, a member of the provincial council, said heavy fighting was continuing in several areas of the city and Afghan forces were in defence mode.
But lawmakers from Ghazni who managed to talk to some residents said the Taleban were in control of much of the city after launching an initial attack in the early hours of Friday (Aug 10).
More than 100 people have been killed and 133 injured in the ongoing Taliban attack in Afghanistan Ghazni.
Militants destroyed a telecommunications tower outside the city, cutting all cellphone and landline access to Ghazni. Afghan forces went from building to building throughout the city, routing any other Taliban forces and expelling them from Ghazni.
While government security forces in the city of Ghazni repulsed the multipronged attack with the help of US air support, Taliban insurgents remained hunkered down on its outskirts, and some were still holed up in residential areas, according to Interior Ministry deputy spokesman, Nasrat Rahimi.
The Provincial Police Chief Farid Ahmad Mashal also told TOLOnews on Sunday afternoon that no government facilities in the city had fallen to the Taliban. "It is just propaganda by the Taliban", he claimed.
Earlier this year, the US sent more military advisers to Afghanistan.
President Ashraf Ghani is said to be considering a cease-fire offer to the Taliban for the upcoming Muslim holiday of Eid al-Adha, which starts August 21. Islamabad denies it is helping the Taliban.
It would also be the most important strategic victory for the Taliban since they lost control of the country after the 2001 USA invasion.