"The goal is to get these people released unharmed, to get them released peacefully and safely."
Cheon Ho-sun, a South Korean presidential spokesman, said: "The Afghan and US governments ... have a certain level of involvement in this issue."
"We have expectations that the two leaders would have sufficient understanding of our position when they hold a summit."
Meanwhile, a South Korean delegation that hopes to negotiate the release of hostages is set to meet with the Taliban amid tight security.
South Korea's Yonhap news agency cited "informed sources" as saying that, with most of the hostages ill, talks were likely to be held later on Friday.
A venue had not been decided on Friday, an official said, but Afghan police agreed any meeting would be afforded high security.
Qari Yousef Ahmadi, a Taliban spokesman, said earlier that the Taliban was ready to meet Kang Sung-Zu, the South Korean ambassador.
The South Korean government has told the Taliban that there is a limit to what it can do to resolve the stand-off.
But Ahmadi said earlier that South Korean officials had "assured" the group that fighters imprisoned by the Afghan government would be released in exchange for the Christian aid workers.
"The Korean delegation has assured us they have spoken to the Americans and the Korean hostages will be freed in exchange with Taliban prisoners," Ahmadi said.
The Taliban have already killed two male hostages, accusing the Afghan government of not negotiating in good faith and ignoring their demand to release prisoners. It has said it will kill the other hostages if its demands are not met.
The government has refused to release Taliban fighters for fear of encouraging kidnapping and after being criticised over a similar deal in March.
The body of Shim Sung-Min, the second hostage to be killed, arrived at Seoul's Incheon airport on Thursday evening.
Amnesty International said on Friday it had appealed in a phone call to Ahmadi for the South Korean hostages to be freed, warning that holding and killing captives is a war crime.
Irene Khan, secretary-general of the London-based group, said in a statement: "Hostage taking and the killing of hostages are war crimes and their perpetrators must be brought to justice."
Ahmadi told Amnesty that "we are trying to resolve this issue ... acceptably," but did not agree to protect the hostages from harm, the statement said.
The South Korean government has called for "flexibility" in negotiations, but has not publicly called for the prisoners to be exchanged for the hostages.
Eight senior members of South Korea's National Assembly travelled to Washington on Thursday to urge US officials to take an "active and positive" approach to the crisis.
There is a widespread perception that Washington could influence Afghanistan's government, although the US has said it will not negotiate with groups it regards as "terrorists".
Officials in Kabul have denied reports of a possible military operation to release the hostages.
In Seoul on Friday, small protests were held outside some mosques.
Meanwhile, about 24 Christian clergymen held a prayer service outside the US embassy in the South Korean capital, praying for the aid workers and urging the US to accept the Taliban's demands to secure the hostages release.
A statement signed by 96 clergymen urged the US to accept the insurgents' main demand, that Taliban prisoners be freed in exchange for the aid workers, now in their third week in captivity.
The statement said: "The US, which massacred civilians in the name of a war against terrorism. must assume responsibility for the current situation.
"South Korea should pull out all troops dispatched abroad for a vicious war of aggression by the US."
On the other side of a line of riot police, though, 12 demonstrators staged a counter-protest.
"Do not use hostage situation to incite anti-US sentiment," one placard read.