The body of 21-year-old Agnes Wanjiru was found at the Lions Court Hotel in the town of Nanyuki two months after she disappeared in March 2012.
The town is close to the Batuk (British Army Training Unit Kenya) camp.
An initial inquiry was unsuccessful, but a fresh investigation has been launched after an inquest delayed until 2019 found Ms Wanjiru was unlawfully killed, the Sunday Times reported last month.
This weekend, the newspaper reports that a soldier accused of the murder has been named by his comrades.
According to the Sunday Times, the soldier allegedly confessed to the killing, and another soldier reported it to senior officers at the time – but no action was taken.
A post-mortem examination found Ms Wanjiru died as a result of stab wounds to her chest and abdomen.
There was also evidence she had been beaten, although due to the condition of her body it was unclear whether she had been sexually assaulted.
Witnesses told the Sunday Times that Ms Wanjiru, a sex worker, was last seen leaving the hotel’s bar with a British soldier.
Labour’s shadow defence secretary John Healey said: “The details of this young Kenyan woman’s death are dreadful, yet there’s still no action from defence ministers on reports of grave failings by the British military exposed in this case.
“There’s been no MoD-led investigation of the soldiers involved and no inquiry into why the MoD failed to respond when Kenyan detectives asked for help.
“Nine years on, justice must now be done for Agnes and her family.
“The Defence Secretary must take this more seriously. He should pledge the fullest co-operation to Kenyan detectives and launch an inquiry into any possible cover-up from commanding officers, military police or the MoD.
“When our forces serve overseas they stand up for British values and these allegations, if proven, would profoundly betray those values.
“This is another case that raises serious questions about the way crimes are reported, investigated and prosecuted in the military.
“The failure of military justice undermines our relationships with allies and the bonds between those who serve with dedication in our armed forces.”
Tweeting about the Sunday Times story, Labour MP Jess Phillips said: “This is brilliant reporting and a tragic tale which I will be pressing the Defence Secretary for answers about.
“Her name was Agnes Wanjiru and we owe her a debt.”
Ms Phillips added: “I have actually stayed in this Hotel I believe while in Kenya meeting sexually exploited women and men.”
An investigation into Ms Wanjiru’s death stalled when a request by Kenyan police in June 2012 to the British Royal Military Police (RMP) that nine soldiers be questioned apparently went missing.
Detectives are said to have asked the RMP to put 13 questions to the soldiers, including whether any of them had sex with Ms Wanjiru on the night she disappeared.
A photo of the victim was included in the request, as well as a request for DNA samples to be taken from the nine men.
The Sunday Times is reporting that the man who allegedly admitted to the killing was not among the nine men.
An MoD spokesperson said: “In 2012, Special Investigation Branch carried out initial enquiries in Kenya, including providing information about British personnel to the Kenyan Police. No further requests were received at that time.
“Following the conclusion of a Kenyan inquest in 2019, we are aware that the Kenyan authorities are looking into this incident.
“The jurisdiction for this investigation rests with the Kenyan Police, and we are currently in discussions with the Kenyan authorities to determine what support is needed.
“Due to this being subject to an ongoing investigation it would be inappropriate to comment further.”