Taliban spokesman Zabiullah Mujahid claimed responsibility for the attack.
Meanwhile, conflicting reports emerged about the airstrike in the Watapur district of Kunar, a province that lies along the border with Pakistan. The territory is dangerous and difficult to reach. Many Arab and other foreign insurgents are believed to operate there alongside the Afghan Taliban. Some are suspected to have links to al-Qaida.
Kunar province police chief Abdul Habib Sayed Khaili said the airstrike hit a pickup truck carrying women and children in Qoro village soon after three Arab and three Afghan militants boarded it Saturday evening. He said some residents called it a drone strike, which would not be unusual in that area.
The police official put the total dead at 15, saying four were women, four were children and one was the civilian driver. But Afghan President Hamid Karzai's office later put the death toll at 16, saying only that women and children were among the victims.
As he has in the past, Karzai strongly condemned the killing of civilians and offered condolences to their families.
NATO spokeswoman 1st Lt. AnnMarie Annicelli said the military alliance carried out a "precision strike" that killed 10 "enemy forces," but that it had received no reports of any civilians dying. Annicelli would not comment on whether a drone was used, but said NATO was still investigating the matter.
"We take all allegations of civilian casualties seriously," NATO said in a statement.
Even as U.S.-led foreign forces draw down their presence in Afghanistan, with a full exit expected by the end of 2014, the air support they provide Afghan troops in many regions is still a crucial part of operations against the Taliban, the resurgent militant movement that wants to topple the U.S.-backed Afghan government.
Past strikes that killed civilians have infuriated Afghans. Karzai even banned Afghan troops from requesting NATO airstrikes during operations in residential areas, though it's unclear how often that ban is enforced.
Tensions also arose Sunday between the U.S. and Afghan governments over a remark by the U.S. special representative for Afghanistan and Pakistan.
In a recent interview with Voice of America's Deewa news service, James Dobbins said, "Well, there already is, of course, a civil war in Afghanistan." Karzai spokesman Aimal Faizi issued a rebuke, saying that if Dobbins' assertion was true, then the U.S. had been an actor in a civil war instead of fighting a battle against terrorism.
In what it called a "policy clarification" posted on its website, the U.S. Embassy in Kabul said the remarks "were not intended to reflect all of the sources of conflict in Afghanistan and should be clarified as in the context of the need for Afghans to speak to Afghans in order to have a lasting peace."