The United States and Afghanistan made progress this week in resolving a dispute over the handling of detainees held in Afghanistan, improving the odds, Afghan President Hamid Karzai said Friday, that the two countries will be able to reach a post-2014 security agreement.

Any such deal, however, could stir concerns in Congress, where lawmakers have objected to giving the Afghans control of captured militants they fear will simply be freed to once again take up their attacks NATO troops.

Speaking to reporters after his meeting with President Barack Obama on Friday at the White House, Karzai said the men had agreed on a “complete return of detention centers and detainees to Afghan sovereignty,” a point of contention in recent months.

Obama did not refer to any such agreement either in his public remarks or in the joint statement that the White House released.

The United States and Afghanistan signed a memorandum of understanding in March, which set a six-month timeline for full transfer of authority of the Parwan detention center at the U.S.-controlled Bagram Air Base to the Afghans.

“This is a sound and solid approach to detainee transfers,” Sen. Lindsey Graham, R-S.C., said in a written statement released at the time. “The agreement enhances Afghan sovereignty but with a robust check and balance system to protect Coalition Forces and continue to apply pressure to the insurgency.”

The agreement, however, reportedly left undetermined the issue of new detainees captured after it was signed, and the United States has since sought to maintain control of those prisoners.

Karzai has complained that in doing so, the United States is violating its pledge.

The two presidents appear to have resolved that disagreement in their meeting Friday, at least according to Karzai.

“With those issues resolved, as we did today … I can go to the Afghan people and argue for immunity for U.S. troops in Afghanistan in a way that Afghan sovereignty will not be compromised, in a way that Afghan law will not be compromised,” Karzai said.

Immunity from local laws was one of the key sticking points between the United States and Iraq when those two countries were trying to negotiate a similar security agreement as the U.S. presence in Iraq was winding down in 2011. Ultimately, the talks collapsed and the Obama administration withdrew all its troops at the end of 2011.

Administration officials have raised the specter of a similar scenario in Afghanistan, should the Afghan government also refuse to grant immunity to any U.S. troops staying after the 2014 deadline for NATO troops’ withdrawal.

On Friday, Obama said that the United States and its NATO allies were again moving up the timing for the transition from NATO to Afghan security control, from summer 2013 to spring of this year. That’s far speedier than the 2014 transition once outlined.

“Our troops will continue to fight alongside Afghans when needed, but let me say it as plainly as I can, starting this spring, our troops will have a different mission: training, advising, assisting Afghan forces,” the president said.

That announcement drew a cheer from Sen. Kirsten Gillibrand, D-N.Y., a member of the Armed Services Committee. “President Obama’s announcement today regarding his policy winding down the war in Afghanistan was a step forward,” Gillibrand said in a statement. “I urge the President to use this momentum to speed up and increase the drawdown of U.S. combat troops.”

Obama said Friday that any follow-on forces staying after 2014 will have two missions: training and assisting Afghan forces and targeted counterterrorism missions against al-Qaida and its affiliates.

Negotiations on the security agreement to formalize that presence are ongoing, the president said, and “will focus on how best to achieve these two tasks.”

“It’s our hope that we can reach an agreement this year,” Obama added.

The White House is reportedly mulling options of keeping 3,000; 6,000; or 9,000 troops in Afghanistan after 2014, far smaller figures than the up to 20,000 forces the U.S. military originally envisioned. A White House official said this week that the option of removing all troops would also be discussed.

Some Republicans on Capitol Hill, including Graham and Sen. John McCain, R-Ariz., have argued that a larger force is needed to shore up the Afghan army and combat al-Qaida. The vast majority of congressional Democrats, however, would prefer as small a military footprint as possible, or no force at all.

“I stand ready to work with the President and the Department of Defense on solutions to continuing U.S. security in South East Asia that does not involve the presence of U.S. combat troops in Afghanistan,” Gillibrand said in her statement.