A senior Afghan Defence Ministry official told Reuters and CNN that Fazlullah was targeted and killed.
The strike comes amid a seven-day ceasefire between the Afghan Taliban and government security forces to allow Afghan citizens to observe the last days of Ramazan and Eid peacefully.
Nicholson said the United States would adhere to the unilateral cease-fire announced by Ghani, which did not apply to US counterterrorism attacks on other terror groups.
The fearless Swat girl was later awarded the Nobel Peace Prize for her efforts for girls education and facing the grave threats posed to her by the Taliban.
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Border Patrol agents stopped two of the vehicles and made multiple arrests, but the third vehicle continued, the statement said. The Trump administration has said tougher immigration policies are needed to deter immigrants from entering America illegally.
The US State Department says the TTP has "demonstrated a close alliance with Al-Qaeda" and had given explosives training to Faisal Shahzad, the would-be Times Square bomber.
He is also believed to have ordered the 2012 shooting of then-15-year-old Yousafzai over her advocacy of girls' education.
Unconfirmed reports from locals said Fazlullah has been killed, the Voice of America reported.
The High Peace Council has said that the Afghan government is ready to extend the ceasefire if the Taliban gives a green light. The killing of TTP leader is likely to ease prevailing tension between the U.S. and Pakistan. He said the USA drone targeted a vehicle in which Fazlullah and the other militants were riding.
Three army soldiers were martyred and five militants killed in attacks from across the Pak-Afghan border on military posts in North Waziristan district on Friday, sources said.
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West Texas Intermediate (WTI) crude oil futures were up 0.2 per cent at US$67.02 per barrel, while Brent was at US$76.02. Mr Draghi made the announcement at a news conference following the bank's meeting to discuss eurozone monetary policy.
Meanwhile, U.S. officials said on Friday that Fazlullah was targeted in Thursday's strike in Kunar. "We heard from our people that ameer saib (Mullah Fazlullah) was travelling in a vehicle along with four other commanders when he came under drone attack in Chaugam area of Shegal", said a TTP member on condition of anonymity.
His insurgent group, the Tehrik-e-Taliban, also took responsibility for the brutal attack on an Army Public School in Pakistan's north-western city of Peshawar in December 2014, when more than 140 children and their teachers were slaughtered.
Fazlullah had been a major figure in the TTP even before he became emir in late 2013, and led a Pakistan Taliban militia in the country's Swat Valley prior to his elevation to leadership of the group.
In his meetings with Afghan President Ashraf Ghani and U.S. General John Nicholson, who commands American forces and NATO's non-combatant Resolute Support mission, Bajwa discussed bilateral and counterterrorism cooperation.
The Taliban group on Saturday announced that the leadership of the group has chose to declare a ceasefire on three days of Eid Al-Fitr, an announcement which was apparently made in response to President Ghani's ceasefire declaration late last week.
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Pakistan has always been accused of supporting the Afghan Taliban and providing safe haven to its leaders - charges Islamabad denies. "People in Swat will feel safer after the killing of Fazlullah".