Bays also said that Afghan and US troops were "still massed in the area" where the hostages were thought to be held.
Three previous deadlines for the hostages' lives have already passed without consequences.
The South Koreans, a group of Christian evangelicals who had been involved in missionary and aid work in Afghanistan, were kidnapped last Thursday while travelling by bus through Ghazni province.
South Korean negotiators travelled to Ghazni to take part in the negotiations.
The Taliban is, according to Yousuf, demanding eight Taliban members held in Afghan jails to be freed in return for the same number of hostages.
"We had assurance from the Koreans that Kabul will release Taliban prisoners in batches and we will reciprocate," Yousuf said.
The Taliban would then provide further lists of imprisoned fighters, he said.
A prisoner exchange, though, would run counter to a pledge by Hamid Karzai, the Afghan president, not to allow another exchange after his government freed five fighters in March in exchange for an Italian journalist.