Taliban set a "final deadline" of 20:30 GMT on Wednesday for their demands for a prisoner swap to be met.
Seoul said it was urgently checking the claim by the group amid reports that eight of the Christian aid workers had been released.
The hostages were to be moved to a safe zone and then flown back to South Korea after a medical check-up, according to South Korea's Yonhap news agency.
Melissa Chan, reporting for Al Jazeera from South Korea on Wednesday, said that the media in South Korea was broadcasting the report that eight hostages had been moved to safety when it was announced that the hostage had been killed.
The Taliban had threatened to kill the hostages unless Afghan authorities agreed to demands to release prisoners in exchange.
Three previous deadlines for the hostages' lives had previously passed without consequences.
The South Koreans are a group of Christian evangelicals who had been involved in missionary and aid work in Afghanistan.
They were kidnapped last Thursday while travelling by bus through Ghazni province.
The Taliban meanwhile freed a German journalist, the third German to be kidnapped in a week in war-torn Afghanistan, and his Afghan translator hours after abducting them in the east of the country, officials said.
The fresh hostage crisis for Germany was defused after a few hours in eastern Afghanistan's Kunar province bordering Pakistan.
The pair were abducted from a house in the far-off district of Watapour where a NATO-led air strike killed several Afghan civilians two weeks ago, Shalizai Didar, the provincial governor, said.
"They were both freed with the mediation of tribal elders and other influential people. They are safe and sound," Didar told AFP.
The Taliban had earlier claimed responsibility for kidnapping the German journalist.
The two other Germans were kidnapped last week in the southern province of Wardak.
The bullet-riddled body of one was dumped by a road on Sunday while the Taliban said the other was very sick and slipping in and out of consciousness due to diabetes.