Officials say these residual troops would train Afghan forces and help with security.
Karzai on January 10 met with U.S. Secretary of State Hillary Clinton and Defense Secretary Leon Panetta.
After welcoming Karzai to the Pentagon, Panetta said the United States and Afghanistan had arrived at the “last chapter” of the mission to establish a sovereign Afghanistan that can govern and secure itself.
Officials said the meeting between Karzai and Clinton was expected to focus on reconciliation talks with the Taliban and preparations for Afghanistan's 2014 elections.
After a military ceremony in Karzai’s honor outside the Pentagon, Panetta said more than 10 years of war in the aftermath of the September 11, 2001, terrorist attacks on America had helped pave the way for Afghanistan to start standing on its own.
"After a long and difficult path, we finally are, I believe, at the last chapter of establishing an Afghanistan, a sovereign Afghanistan, that can govern and secure itself for the future," Panetta said.
Panetta offered assurances that the United States would continue to be a partner with Afghanistan, even as Washington and its NATO allies withdraw their combat forces.
Karzai expressed confidence that, with continued assistance, Afghanistan would be able to defend itself from Taliban and Al-Qaeda militants.
"I can assure you, Mr. Secretary, that Afghanistan will, with the help that you will provide, be able to provide security to its people and to protect its borders so Afghanistan will not ever again be threatened by terrorists from across our borders," Karzai said.
Panetta later said his talks with Karzai had made “good progress” -- but he did not reveal whether the sides had moved closer to an agreement on the presence of U.S. forces in Afghanistan after 2014.
"It's fair to say that we made some very good progress on all of the key issues that we discussed," Panetta said. "Our meeting, I believe, will help lay the groundwork for President Karzai's discussions tomorrow with President Obama."
Ahead of Karzai’s visit this week, the Obama administration said it has not ruled out a complete withdrawal of all U.S. troops from Afghanistan -- a move that would leave no trainers nor U.S. air support for Afghan security forces.
Officials said this would be a key subject in Karzai’s talks with Obama.
U.S. reports have suggested the White House is considering keeping between 6,000 and 15,000 American troops in Afghanistan after 2014 to continue to pressure militants and train and assist Afghan security forces.
There are currently around 65,000 U.S. troops in Afghanistan, well down from a high of around 100,000, along with some 30,000 troops from other NATO member countries and non-NATO countries.
Karzai’s visit to Washington comes amid some optimism about potential progress in reconciliation talks with Taliban militants.
Members of the Afghan government, the Taliban, and other factions recently met in France for informal talks on easing the conflict.
Neighboring Pakistan has also indicated a renewed willingness to contribute to bringing stability to Afghanistan.