At least 25 people have been killed in a suicide bombing in eastern Afghanistan as elsewhere across the country jubilant scenes marked the first ceasefire between the Taliban and Afghan government in 17 years.
Ghani said in his televised speech that he was ready to discuss the removal of foreign forces, which were excluded from the Taliban's ceasefire and remain a key stumbling block to peace in the eyes of many fighters.
An offshoot of the Islamic State, which has had a foothold in the province and has been behind some of the deadliest recent attacks there, claimed responsibility for the attack outside the provincial capital of Jalalabad, according to the tracking group Terror Monitor.
President Ashraf Ghani said on Saturday he would extend the government ceasefire beyond June 20 and on Sunday the presidential office said it would be extended by 10 days. But over the past two days Taliban fighters could be seen celebrating the truce alongside Afghan troops and other people in a number of locations.
Resident Mohammad Amir said he saw the Taliban casually entering the city of Kunduz.
"If they extend the ceasefire they will be compelled to talk, which I think the Taliban isn't interested in".
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Afghan Taliban spokesman Zabihullah Mujahid said in a telephone interview that he could not confirm Fazlullah's death because of the remoteness of the area but also because Afghanistan's Taliban are not present in that area.
Ghani also repeated his promise that everything could be on the negotiation table, including the presence of foreign forces.
Meanwhile, videos and photos posted on news sites and social media showed soldiers and Taliban greeting and hugging each other and taking selfies in several provinces.
A Taliban fighter told the AFP news agency that, while he welcomed the ceasefire, long-lasting peace would only be achieved if USA forces left the country.
Amarullah Saleh, the former head of the National Directorate of Security, said Mr Ghani had committed "a grave mistake" by allowing Taliban fighters to enter government-controlled areas.
In eastern Logar province, dozens of Taliban on motorcycles roared through the provincial capital of Pul-e-Alam, some of the vehicles festooned with the Afghan flag.
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Some Afghan citizens have been calling on the Afghan government and the Taliban to extend the unprecedented cease-fire.
The Taliban surprised everyone on Monday by announcing the three-day ceasefire over Eid, which began yesterday.
Thursday's drone strike, which reportedly killed Fazlullah and five other insurgents when missiles slammed into the vehicle in which they were driving, occurred just hours before Afghanistan's Taliban began a three-day cease fire.
"It was the most peaceful Eid". The Afghan government has also released some Taliban militants from prison, the BBC reported. "I felt for the first time in the eight years of my jihadist life that I am at least safe".
Ghani in his statement announcing the extension called on the Taliban to reply in kind. "It is hard to describe the joy", said Qais Liwal, a student in Zabul.
"We are all just so exhausted of war", he said.
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