"The Fisher House Foundation will provide the families of the fallen with the benefits they so richly deserve," Defense Secretary Chuck Hagel said in a statement, adding that the Pentagon would reimburse the foundation after the shutdown ended.
Hagel said Fisher House, which works with veterans and their families, had approached the Pentagon about making the payments. The Defense Department typically pays families about $100,000 within three days of a service member's death, but officials say the shutdown was preventing those benefits from being paid.
A senior defense official said the government could not actively solicit funds from private organizations but could accept an offer. The official was not authorized to publicly discuss the offer by name and insisted on anonymity.
The failure to make the payments has stirred outrage on Capitol Hill and at the White House. Obama spokesman Jay Carney said Wednesday that the president was "disturbed" when he found out the death benefits had been suspended and demanded an immediate solution.
"The commander in chief, when he found out that this was not addressed, he directed that a solution be found, and we expect one today," Carney said before the Pentagon announced the agreement with Fisher House.
The Republican-led House unanimously passed legislation Wednesday to restore the death benefits. But it's unclear whether the Democratic-led Senate will take up the measure or whether Obama would sign it. Obama has threatened to veto other legislation passed by the House in recent days that would reopen individual funding steams, arguing that a piecemeal approach to ending the shutdown was unacceptable and that the entire government must be reopened.
As of Wednesday afternoon, the Obama administration had yet to issue a formal veto threat for the death benefit bill.
Before the government shutdown last week, Congress passed and Obama signed a bill allowing the military to be paid during the federal closure. However, the death benefit payments were not covered by that legislation.
Carney said the Pentagon told lawmakers before the shutdown that the death benefit payments were not covered by the bill and would be cut off during a shutdown. However, he repeatedly refused to say when the president was first told that death benefits would not be paid.
Amid the controversy, Hagel made a rare trip Wednesday to Dover Air Force Base for the arrival of remains of four soldiers killed in Afghanistan. The remains of every U.S. military member killed overseas are flown to Dover for processing. Family members attend the arrival, but the secretary of defense usually does not.
Among the soldiers whose remains were brought to Dover on Wednesday was Pfc. Cody J. Patterson, 24, of Philomath, Ore. Patterson's family allowed members of the media to witness the return of his remains, but an Army liaison officer who works with mortuary officials at Dover said the family did not want to talk to reporters.
The other soldiers whose remains were returned were 1st Lt. Jennifer M. Moreno, 25, of San Diego, Calif.; Sgt. Patrick C. Hawkins, 25, of Carlisle, Pa.; and Sgt. Joseph M. Peters, 24, of Springfield, Mo. In a statement Tuesday, U.S. Army Special Operations Command said Hawkins and Patterson were members of the 3rd Battalion, 75th Ranger Regiment, and Peters was a special agent assigned to the 286th Military Police Detachment.
Hagel put his hand over his heart as white-gloved soldiers carried the flag-draped case carrying Patterson's remains from a C-17 cargo plane to a white panel truck for transfer to the Dover mortuary.
Associated Press writers Bradley Klapper in Washington and Randall Chase in Dover, Del., contributed to this report.