Afghan MPs have demanded that the man be tried in Afghanistan, but the scenario is very unlikely.
Meanwhile an Afghan man who crashed a lorry at an airfield as the US defense secretary’s plane was arriving has died of his injuries, officials say.
US Defense Secretary Leon Panetta was not at risk at any time, US officials said.
Lt. Gen. Curtis Scaparrotti said the man had apparently tried to ram the stolen vehicle into a group of US Marines at Camp Bastion in Helmand.
Leon Panetta was in Helmand to address US troops, as fears mount that they could be the target of a backlash against foreign forces.
He also met Afghan President Hamid Karzai in an effort to rebuild relations rocked by incidents such as the massacre in Kandahar and the burnings of Korans at a US military base last month.
NATO has insisted that the detained man carried out the killings on his own.
But the head of an Afghan parliamentary delegation said he has heard evidence from local villagers which suggests as many as 20 US soldiers were involved.
Sayed Ishaq Gillani, a leading Afghan MP, also claims that helicopters were heard overhead, and that they were seen dropping chaff – a measure designed to protect aircraft from ground attack.
Sayed Ishaq Gillani said local people believe the killings were carried out in revenge for an attack a week earlier in which several US troops were hurt.
The victims were shot in their homes in the remote Panjwai district of Kandahar, which is also the spiritual homeland of the Taliban.
According to the NATO version of events, the staff sergeant, who has not been named or charged, allegedly left his base in southern Afghanistan before dawn on Sunday, entered several houses in the area and shot men, women and children at close range.
“We do not have appropriate detention facilities in Afghanistan,” Captain John Kirby said.
A NATO official later confirmed that the suspect had been flown to Kuwait.
Members of the Afghan parliament had demanded that he should be put on trial in their country.
But this was never going to happen. The US has always insisted that charges of wrongdoing by its soldiers be dealt with within the American military legal system.
US officials say the soldier handed himself in. Leon Panetta has said that if found guilty, he could face the death penalty.
Officials said the soldier had completed several tours in Iraq but was on his first tour of duty in Afghanistan.
NATO and the US administration have insisted that there will be no change of strategy in Afghanistan. The issue has been at the top of the agenda as the UK’s Prime Minister David Cameron makes a state visit to Washington.
NATO’s International Security Assistance Force (ISAF) plans to withdraw all of its combat forces by the end of 2014. American troops are also following that timetable.
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