A spokesman for the provincial governor of Nangarhar confirmed a vehicle bomb was responsible for Saturday's attack in the town of Ghazi Aminullah Khan.
A US official said that the USA believes that it is likely the strike killed Fazlullah, but efforts are ongoing to confirm his death.
In a rare televised address, Afghan President Ashraf Ghani announced on Saturday the government was extending the week-long ceasefire which was due to end on Tuesday.
"I confirm that Mullah Fazlullah, leader of the Pakistani Taliban, has been killed in an joint air operation [with the US] in the border area of Marawera district of Kunar province", Mohammad Radmanish, spokesman for Afghan defence ministry, told Reuters news agency, adding the air attack was carried out at about 9am local time on Thursday.
Earlier in June, the Taliban made a surprising statement announcing the suspension of hostilities with the government forces for three days to mark the Muslim holiday of Eid.
The Pakistan government statement also said an "action had finally been taken against an enemy of the people and state of Pakistan".
His announcement comes as Taliban fighters and Afghan security forces hugged and took selfies with each other across the country as an unprecedented ceasefire in the war-torn country held for the second day of Eid.
Before the Taliban's ceasefire had even started, analysts had expressed cautious optimism that the truces, if successful, could help build trust between the government and the Taliban and lay the groundwork for peace talks.
Despite reports that the Taliban's ceasefire might be prolonged, Taliban spokesman Zabiullah Mojahid told CNN, "we have no intention to extend that".
The group has killed hundreds of people in multiple attacks across the country since it first emerged in the region in 2014.
Zabihullah Mujahid also said the group banned its fighters from attending public gatherings, going into cities, and meeting with the Afghan public, as well as officials and security forces, during the cease-fire. They're looking for an outright victory, ' one diplomat told AFP on the condition of anonymity.
In many cities, Taliban fighters left their weapons at the gates when they went to Eid celebrations.
"Ghani has full blessings of the USA administration and it is crucial for the US officials to prove that Trump's policies are working and talks with the Afghan Taliban are imminent", said a senior diplomat who met Afghan officials on Sunday to monitor the chance of back-channel talks with Taliban leaders.
"We are calling on the Taliban, they saw in the media that in villages and districts and all parts of the country people called for the ceasefire to be extended extending and for directs talks".
The government has also released some Taliban militants from jail.
In a statement later Saturday, the USA government and North Atlantic Treaty Organization forces in Afghanistan supported Mr. Ghani's extension of the cease-fire.
After the attack, Afghan President Ashraf Ghani announced a nine-day extension of the cease-fire.
US Secretary of State Mike Pompeo said Saturday that the United States supports a move by the Afghan leader to extend the ceasefire.
Taliban fighters ride their motorbikes inside Ghazni city, capital of Ghazni province, west of Kabul, Afghanistan, Saturday, June 16, 2018.
"The U.S.is prepared to support, facilitate and participate in these discussions", he added.
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