Two male hostages were killed in late July and two women were released earlier this month.
Another 14 women and five men are still being held captive.
Yousuf Ahmadi, speaking by telephone from an unknown location, gave no further details of the direct talks, except to say that it will start at 0530 GMT on Tuesday.
The Taliban has threatened to kill more hostages unless Afghan authorities agree to release jailed fighters in exchange.
They have also demanded that South Korea withdraw its troops from the country.
Seoul has said it would pull out its 200 troops, mostly engineers and medical workers, by the end of the year as scheduled.
On Monday a South Korean defence ministry spokesman denied that the move to withdraw troops came after an agreement with the Taliban as reported by Afghan state media.
The Afghan Islamic Press news agency said the Taliban had agreed to release all 19 hostages upon the withdrawal of the South Korean military contingent from the Bagram base.
But South Korean government officials said on Sunday no agreements have been reached over the hostage situation.