US-backed Afghan government forces and Taliban fighters clashed as the troops tried to gain access to the wreckage of a US military plane.
The security forces were reportedly ambushed by the Taliban, which controls the territory where the jet crashed in central Ghazni province, a police chief claims.
A Taliban spokesman claims Afghan forces backed by US military support had tried to capture the area around the crashed aircraft, but were repelled by fighters from the Islamist militant group.
The US has disputed the Taliban's claims that it shot down the plane, with one American defence official claiming the aircraft suffered a mechanical error.
Afghan and foreign troops were planning to conduct an operation to access the wreckage in snow-covered Deh Yak district, where the plane went down on Monday afternoon.
The Taliban hasn't provided any evidence to back up its claims that it downed the Bombardier E-11A, which had US Air Force and Air Combat Command logos, in the remote and mountainous region.
There were conflicting claims over the number of dead and people on board.
The Taliban said there were no survivors, it had recovered six bodies and the victims were CIA officers.
An Afghan police chief said four were dead and two were missing.
A US official told Reuters the aircraft was carrying fewer than five people.
The remains of passengers and/or crew, the charred wreckage and anything salvaged from the aircraft, which burst into flames, are now in Taliban hands.
Photos posted on social media allegedly show charred bodies and documents recovered from the wreck.
Afghan special forces, rescuers and air crash investigators have been unable to access the site.
Ghazni provincial police chief Khalid Wardak told Reuters that security forces were ambushed by the Taliban on Tuesday.
He added: "As per our information, there are four bodies and two on board were alive and they are missing."
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The Afghan troops were given an order to retreat, he said, adding that airborne action is to be taken instead.
Taliban spokesman Zabiullah Mujahid told Reuters that Afghan forces backed by US military support had tried to capture the area around the crashed aircraft and clashed with fighters of the Islamist militant group.
The attempt was repelled, he said.
He said the Taliban would allow a rescue team access to recover bodies from the crash site.
He added: "Taliban fighters on the ground counted six bodies at the site of the US aeroplane crash."
The Taliban spokesman said there could have been more victims, but the militant group could not be certain because fire had reduced everything to ashes.
Speaking on condition of anonymity, US officials said the plane was carrying fewer than five when it crashed, with one official saying initial information showed there were at least two.
The site has not been visited by US officials or any other members of the international force in Afghanistan, but the Taliban claim to have brought down the plane is misleading, a US defence official told Reuters.
The official said a preliminary probe showed there was a mechanical error.
The crashed aircraft, built by Bombardier Inc, is used to provide communication capabilities in remote locations.
Earlier, US Forces Afghanistan spokesman Col Sonny Leggett said: "A US Bombardier E-11A crashed today [Monday] in Ghazni province, Afghanistan.
"While the cause of crash is under investigation, there are no indications the crash was caused by enemy fire.
"We will provide additional information as it becomes available.
"Taliban claims that additional aircraft have crashed are false."
Meanwhile, Iranian state TV claimed a top CIA chief who orchestrated the assassination of an Iranian general was killed in the crash, without providing any evidence.
The US has not yet commented on claims that Michael D’Andrea, who was involved in the hunt for Osama bin Laden and is nicknamed the "Dark Prince", was on board the jet and among those killed.
D'Andrea, the head of CIA operations against Iran, Afghanistan and Iraq, is said to have planned America's killing of Iranian General Qassem Soleimani, who was killed in a US drone strike in Baghdad earlier this month.
In a report on Tuesday, Iranian state media outlet Mizan, the mouthpiece for Tehran's Judiciary, claimed D'Andrea was among the dead.
The Veterans Today website, which has been accused of spreading Russian propaganda and reportedly had ties to Iran’s state-backed PressTV in the past, also claimed D'Andrea was killed, citing unnamed Russian intelligence sources.