The raids targeting Taliban militants were reportedly carried out on October 24 in the Panjwayi and Pashmul districts of Kandahar.
A local man told RFE/RL's Radio Free Afghanistan that a nomad camp outside a village was hit during one of those strikes, killing and injuring him and other noncombatants.
The man said he lost many relatives in the bombing.
"We live in the desert; we are nomads," the man said. "There is only one house in that area, and we had our tents near it; they bombed us at night. Nineteen members of my family were killed; the number of dead is more than 60."
The man, who was being treated at a hospital in Kandahar, called the NATO strike "unjustified" and insisted that there were no Taliban in the area.
Local officials estimate the number of civilian deaths at between 50 and 90. They say explosions destroyed a number of houses and killed villagers. Locals say about 20 survivors of the strikes are being treated at Kandahar's main hospital.
A member of the Panjwayi district council, Haji Nik Mohammad Khan, told Radio Free Afghanistan that he helped recover bodies buried under the rubble of bombed-out homes.
"As of [October 25], there were 61 killed; I prepared the list of dead myself," Mohammad Khan said. "[The next day] the number of dead increased to 63. We initially estimated the number of injured at 30 to 35. But after visiting the hospital, we discovered that 20 were injured."
Mohammad Khan said women and children are among the dead. He, too, insisted that there were no Taliban in the area.
NATO officials said today that a preliminary review suggested that 12 civilians were killed in three battles in the Panjwayi district of the former Taliban stronghold of Kandahar. NATO has pledged to cooperate fully with the Afghan Defense Ministry as investigations continue.
News agencies initially quoted a spokesman for NATO's International Security Assistance Force (ISAF) suggesting that as many as 70 militants were killed in the clashes in Panjwayi. The spokesman, Major Luke Knittig, today told RFE/RL that 48 militants had died when ISAF troops targeted militants with artillery fire and air strikes. He also stressed that the alliance regretted any civilian casualties.
"Sadly it is possible that there are more [civilian casualties]," Knittig said. "But I don't think it's accurate that there were strikes on Panjwayi -- what it was is us defending the four bases [that] we maintain in those two districts. Since defeating the Taliban there a month and a half ago...we had three different attacks on our bases there. Ultimately, we used some air [strikes] to defend ourselves from the attack on our base there, and very sadly, some civilians were killed."
NATO has frequently accused the Taliban of using civilians as human shields.
Bismallah Afghanmal, a provincial council member, has also said that fighters fled into civilian homes that were then attacked by NATO forces.
Afghans 'Hurt And Saddened'
President Hamid Karzai responded with a statement saying he has been "deeply touched" by reports that civilians were been killed or injured in ISAF bombardments in Kandahar.
"The people of Afghanistan are hurt and saddened by this [NATO] incident," Karzai told a news conference today in Kabul. "We are sad, [and] we share our solidarity with the family of victims. I have spoken with 75-year-old tribal leader, Haji Abdullah Shah. He was crying when I was talking on the phone."
Karzai has appointed a delegation to travel to the area and investigate.
Spokesman Knittig told RFE/RL that ISAF will await those findings.
"We're going to listen very closely to what the delegation [that] President Karzai sent down finds," Knittig said. "But the fact that we had insurgents firing at us with rocket-propelled grenades and machine guns, and formed groups actively trying to overrun our bases, gives us a lot of confidence that when we say we killed 48 insurgents, we did."
Karzai in mid-October urged international forces to exercise greater care to avoid civilian casualties after up to 20 civilians were reported killed in two military operations.
His message at today's press conference was that Afghanistan must bolster domestic security forces but that it also needs international support to counter terrorism.
"We need to strengthen our Afghan forces," Karzai said. "At the same time, as I have mentioned several times, the international community must find the source of terrorism, where [the terrorists] are financed and equipped and where they get training. We should eliminate that."
The civilian casualties have angered the public, and ISAF's British commander, British Lieutenant General David Richards, estimated on October 8 that two in three Afghans might soon back the militants' cause if there is no visible improvement in their lives.
On October 26, the UN mission in Afghanistan (UNAMA) expressed serious concern over reports of civilian deaths in Kandahar. The UN said the safety and welfare of civilians must always be paramount and added that any civilian casualties are "unacceptable."
Hundreds of people -- many of them guerrillas but also civilians -- have been killed in Afghanistan during the recent escalation of violence that has been blamed on militants.
Today, there was more violence. A bomb blast killed at least 14 people in the city of Tirin Kot, the capital of Afghanistan's southern Oruzgan Province. That explosion reportedly tore apart a minibus full of civilians -- some of them children. No one has claimed responsibility for that attack.
(Radio Free Afghanistan correspondents Javid Ahmad Wafa in Kandahar, Ahmad Takal in Prague, and RFE/RL Prague correspondent Golnaz Esfandiari all contributed to this report)