U.S. Secretary of State John Kerry and Afghan President Hamid Karzai say they are closer to an agreement on bilateral security but differences remain over the question of immunity for U.S. troops.
The two officials commented in Kabul late Saturday, following extended talks on a possible U.S. presence in Afghanistan after the NATO-led military mission ends next year.
President Karzai said the question of which country had jurisdiction over any crimes committed by U.S. forces after 2014 would have to be resolved by the Afghan parliament and an assembly of elders.
Kerry said the U.S. could prosecute any wrongdoing by American forces.
The U.S. has been pushing for an agreement with Afghanistan by the end of this month.
The talks between Washington and Kabul have been in progress for more than a year.
However, they have repeatedly stalled on issues of Afghan sovereignty, the expected role of U.S. forces, and Karzai's demand for American guarantees against future foreign intervention. If there is no agreement, U.S. forces will leave by the end of 2014.
In another development, U.S. officials announced the capture of senior Pakistani Taliban commander Latif Mehsud.
Word of his capture came as Kerry met with Karzai on Friday.
A U.S. State Department spokeswoman said Mehsud was captured in eastern Afghanistan.
The spokeswoman said Mehsud's group Tehreek-e-Taliban Pakistan had claimed responsibility for attacks that include an attempted bombing in New York's Times Square in 2010.
Some information for this report was provided by AP and AFP.