In a sign of the sensitivity of the issue, Karzai twice described the final step of the process as consultations, then changed his wording to "an exchange of views." He said that "I must use my words carefully."
"We will consider it," he said of a U.S. request to keep particular prisoners in custody, adding that Afghan law would call for such handling of anyone posing a legitimate threat.
The matter touches on a politically charged situation for Karzai, who is criticized by the Taliban and other foes for the continuing presence of the NATO forces in the country.
The handover deal was reached Saturday, but came a full year after the initial plan was announced by U.S and Afghan authorities.
An initial plan called for the prison and detainees to be handed over within six months, but in September, the United States "paused" the transfer of detainees, which include suspected Taliban militants and insurgents.
Other tensions between the countries involve continuing violence, such as a bomb blast in Kabul this month that killed nine people. Karzai said afterward that there are "ongoing daily talks between the Taliban, Americans and foreigners in Europe and in the Gulf states."
The comment effectively claimed the United States was trying to foment continued violence inside Afghanistan, and it was quickly denounced by NATO and U.S. officials.
Some experts say Karzai's comments were fueled by frustration over the detention facility not being handed over sooner, which he viewed as an attack on his country's sovereignty.
During the news conference, Kerry repeatedly referred to Afghanistan's sovereignty in the transition process as the NATO military force turns over leadership of combat operations to Afghanistan's military, which it has trained in recent years.
Under an agreement between the Afghan government and NATO, the bulk of U.S. and NATO combat forces are to be out of Afghanistan by the end of 2014. What remains to be decided is how many troops may remain to help train Afghan forces.
The trip was Kerry's first visit to Afghanistan as secretary of state, but the former senator's sixth during Obama's presidency.
He was in Iraq over the weekend to press that nation's leaders to take steps prohibiting Iranian planes from delivering arms to Syria's besieged government. Last week, Kerry joined Obama on a trip to Israel, the Palestinian territories and Jordan.