In apparent contradiction to what the Afghan government has said, Bashir claimed that Kabul will meet their demands and that the Korean and Afghan sides had already approved an initial "prisoners exchange".
Speaking at the Afghan Red Cross office where the second day of talks were being held, Bashir said: "We have great hope that the hostage crisis will resolved today or tomorrow inshallah [God willing]."
The Taliban negotiator also said the hostages, mostly women, were safe and healthy, and that he was "optimistic about the result of our talks".
The talks between the Taliban leaders and four Korean officials, which began on Friday, was the first face-to-face meeting since the Koreans were kidnapped.
Qari Yousef Ahmadi, a Taliban spokesman, told Al Jazeera that its officials had received a written guarantee from the Afghan government that they would not be arrested at the meeting.
The meeting comes after days of stalled negotiations on the safe release of the hostages.
"This is a very hopeful development in this situation," Nolan said.
"We had gone almost a week where they have been looking for an appropriate meeting venue to hold face-to-face negotiations."
Twenty-three South Koreans, who were working with a church group as health aid volunteers, were abducted on July 19 as they travelled by bus from Kabul to the southern city of Kandahar.
Two male hostages have been executed by the Taliban since then.
Call for pull out
South Korea's government has also called for aid organisations to leave Afghanistan by the end of the month, citing safety reasons, an embassy official told the Associated Press.
Ahmadi said the departure of South Korean aid workers would move forward negotiations with the Taliban.
Cha Sung-min, a spokesman for the hostages' families in South Korea, said the mothers of several female hostages will travel to Dubai next week to seek help from the Arab world.
Source: Al Jazeera and agencies