The private, from 1st Battalion, Royal Gurkha Rifles, was said to have been flown back to the UK and suspended following the incident in the Babaji area of central Helmand earlier this month.
According to the Mail on Sunday, the Gurkhas were told they were hunting a "high value target" and must prove they had killed the right man.
But the soldiers came under heavy fire as they were attempting to remove the Taliban commander's body.
It is understood that the Gurkha then cut off the dead man's head with his kukri knife and took it away from the area so his identity could be verified.
This is considered hugely offensive to Afghan Muslims whose customs dictate that the entire body is buried, even if body parts must be retrieved.
The Ministry of Defence said it was investigating the incident and had informed the Afghan authorities. Under the Geneva Conventions, soldiers are banned from demeaning enemy combatants.
The Gurkha could face a court martial and if found guilty of beheading the Taliban fighter, could be jailed, the paper reported.
Colonel Richard Kemp, a former commander of British forces in Afghanistan, said: "In this case, it appears that the soldier was not acting maliciously, but his actions were clearly ill-judged.
"The Gurkhas are a very fine regiment with a proud tradition of service in the British forces and have fought very bravely in Afghanistan.
"I have no doubt that this behaviour would be as strongly condemned by the other members of that regiment, as it would by all soldiers in the British forces."
A Ministry of Defence spokesman said: "We are aware of an incident and have informed the Afghan authorities.
"An investigation is under way and it would not be appropriate to comment further until this is concluded."