Army Chief of Staff Kwon Oh-Seong offered to step down during a meeting with Defence Minister Han Min-Koo, a ministry spokesman said. It was not immediately clear if the resignation had been accepted.
Kwon expressed regret and apologised for the death of the 23-year-old private, saying he "feels responsible", the spokesman told AFP.
"I hope this kind of incident will never happen again," the army chief was quoted as saying.
President Park Geun-Hye also voiced her sadness over the case and called for severe punishment of those responsible.
The five have been arrested on manslaughter charges after an initial investigation showed the private had been repeatedly bullied.
The case came on the back of two separate suicides by army privates last month, and a deadly shooting spree in June in which a sergeant killed five members of his unit for taunting him.
Investigators found Yoon had been the target of regular bullying and assaults, including sessions of crude water-boarding.
Barrack-room bullying has long tainted South Korea's military service, which is mandatory for all able-bodied men between the ages of 18 and 35.
Conscripts, most of them in their early twenties, account for the lion's share of the military's 690,000 active personnel.
The defence minister had made his own public apology yesterday, describing the death as "an inhumane crime" that shamed a civilised society in the 21st century.
Han said an army division commander would be dismissed from his post and disciplined, together with other officers under his command.
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