Pentagon spokesman George Little declined to comment on the specifics of the deal during a briefing on Tuesday, except to say both sides were "making progress" in the negotiations.
The raids have been "a concern for some time" among American military leaders and their Afghan counterparts, Little said. But commanders on both sides clearly "recognize the effectiveness" of those raids in the ongoing war effort.
It would also require U.S. and coalition forces to get a warrant from the Afghan courts before carrying out any future raid, according to those reports.
Both sides were ready to lock the deal in place on Wednesday, according to the Times.
However, disagreements surfaced over whether terror suspects captured during the raids would be kept under Afghan custody, or remanded to American forces.
On Monday, the ANSF officially took control of all terror detainee operations in Afghanistan from U.S. and NATO commanders.
The night raids, which are usually carried out by U.S. special-operations forces, are currently supported by a mix of Afghan and coalition troops, Little told reporters during Tuesday's DOD briefing.
He could not comment on what the exact force mix was, but did say those support teams were evenly split between American and coalition forces and the ANSF.
But "at the end of they day [Afghans] are responsible" for creating a process where the raids can continue, Little said, especially as U.S. troops prepare to leave the country by 2014.
American and NATO commanders in Afghanistan have long credited the use of night raids as an important tool in taking out key elements of the Afghan insurgency.
Combined with airstrikes by CIA-operated drones, the raids have been credited with capturing or killing many key leaders within the Taliban, al Qaeda and other terror groups fighting in Afghanistan.
Al Qaeda leader Osama bin Laden was killed during a night raid by U.S. special forces on his Abottabad compound in Pakistan last May.
But the raids have infuriated Afghans, who claim innocent civilians have been injured or killed during them.
Some Afghan civilans believe the March 11 shooting spree by Staff Sgt. Robert Bales was part of a night raid, and not the act of a rogue soldier.
A deal could be finalized next month, during NATO's annual summit in Chicago. It could be part of a larger Strategic Partnership Agreement with the Afghan government.
That deal would set the overall terms for a continued U.S. presence in Afghanistan after the 2014 deadline.