President Bush and Afghan President Hamid Karzai say there should be no prisoner swap to gain the freedom of 21 South Korean hostages held by the Taleban in Afghanistan.
During a second day of talks outside Washington, both leaders agreed a prisoner exchange would only embolden Taleban insurgents.
President Bush called for the immediate release of the hostages, who are volunteer aid workers from a South Korean church.
Taleban kidnappers are demanding the release of militants in exchange for the hostages. The kidnappers have already executed two of the South Koreans.
Earlier, a Taleban spokesman told news agencies the lives of the hostages are in the hands of Mr. Bush and Mr. Karzai.
Most of the 21 remaining captives are women, and many are reported to be sick after being held by the Taleban for nearly three weeks.
South Korean officials say the Taleban allowed one of the hostages to speak by phone to diplomats at South Korea's embassy in Kabul on Saturday.
In telephone calls to VOA and other news organizations Saturday, the hostages said two women in the group are seriously ill.
A shipment of medicine intended for the captives was delivered Sunday to a location in southern Afghanistan designated by the Taleban.
Efforts to arrange direct talks between the Taleban and South Korean officials have stalled with no agreement on a location for the negotiations.