South Korea is under intense pressure to bring the hostages home but concedes it has few cards to play. Seoul has called for "flexibility", a comment analysts say is directed at the United States to pressure Kabul to strike a deal with the kidnappers.
"We plead with the U.S. government that it call on the world for its interest and support so that the remaining 21 can return safely through peaceful and humanitarian means, not through the use of force," the family members said in a statement delivered to the U.S. Embassy in Seoul.
"Please help. Please help our families' members," said the statement read to reporters by the mother of one of the hostages.
The families' statement came amid growing calls in South Korea for Washington to intervene, including a joint statement by five political parties representing the largest groups in parliament.
"We, with deference, request an active position and role by the U.S. government and the United Nations in order to prevent imminent killings," they said in a statement.
A U.S. State Department spokesman has said Washington "does not make concessions to terrorists".
Taliban insurgents have shot dead two Korean men of a group of 23 sent by a Christian church in suburban Seoul to do relief work in Afghanistan. They have threatened to kill more if their demands for the release of prisoners is not met.
Separately, South Korea on Thursday officially put Afghanistan on a list of countries to which travel is banned.
