At least two of four US Marines shown in a video appearing to urinate on Taliban corpses have been identified, a Marine Corps official has told the BBC.
The video, which was posted online, purports to show the Marines standing over the bodies of several Taliban fighters, at least one of whom is covered in blood.
The Marines have begun a criminal investigation and an internal inquiry.
US officials and Afghan officials have condemned the video as "deplorable".
The Taliban condemned the video, but said it would not affect the political process.
The origin of the video is not known, but it was originally posted to YouTube. It has not been verified but correspondents say all indications are that it is authentic.
The BBC's Steve Kingstone says the official would not confirm the Marines' whereabouts, but reports suggested the unit involved was based at Camp Lejeune, North Carolina - a major military base.
US media reported that the unit belonged to the 3rd Battalion, 2nd Marine Regiment.
That battalion has been deployed to a wide range of combat and peacekeeping situations, from Iraq, Afghanistan and Guantanamo Bay to fighting wildfires in Idaho.
The unit deployed to Afghanistan in early 2011 and returned in September or October, CNN reported.
A US Marines spokesman, Lt Col Joseph Plenzler, told the AFP news agency that "we cannot release the name of the unit at this time since the incident is being investigated".'Total dismay'
Earlier, US Defence Secretary Leon Panetta condemned the acts shown in the video and vowed that a full investigation would be carried out by the Marines and the Kabul-based International Security Assistance Force (Isaf).
"This conduct is entirely inappropriate for members of the United States military," Mr Panetta said in a statement.
The US defence secretary said he had seen the footage, and the Pentagon confirmed that he had spoken by telephone with Afghan President Hamid Karzai.
Mr Karzai condemned the attacks in strong terms.
"I find the behaviour depicted in it utterly deplorable. This conduct is entirely inappropriate for members of the United States military," he said.
In a separate news conference, Secretary of State Hillary Clinton expressed her "total dismay" at the video.
Mrs Clinton said she shared Mr Panetta's view that such behaviour was inconsistent with the standards "that [the] vast, vast majority of our personnel - particularly our marines - hold themselves to".
The video did not change the nature of US efforts to secure Afghanistan, she said, saying the US continued to support security and reconciliation efforts that were "Afghan-led and Afghan-owned".
Tentative moves are being made towards peace in the country, ahead of the planned withdrawal of international combat forces by the end of 2014.
The US has about 20,000 Marines deployed in Afghanistan, based mostly in Kandahar and Helmand provinces. In total, about 90,000 US troops are on the ground in Afghanistan.
The Taliban said last week that they were working to set up a political office, possibly in Qatar, that would help to facilitate negotiations with the Afghan government and Nato countries.
US special envoy Marc Grossman will go to Qatar and Afghanistan next week in support of further talks.
Taliban spokesman Qari Yousuf Ahmadi told the BBC that this was not the first time Americans had carried out such a "wild action" and that Taliban attacks on the Americans would continue.
But a different Taliban spokesman, Zabihullah Mujahid, said the video "is not a political process, so the video will not harm our talks and prisoner exchange because they are at the preliminary stage".
However Arsala Rahmani, a senior member of the Afghan government's High Peace Council, told Reuters the video would "leave a very, very bad impact on peace efforts".
The Taliban are known for applying a ruthless brand of Islamic Sharia law in areas they control and have carried out many suicide bombings and attacks which have killed civilians.