Barack Obama, the US president, signed an order to eliminate so-called black sites run by the Central Intelligence Agency (CIA) in January, but the Bagram facility is operated by military special operations forces.
Military officials said earlier this month that suspected high-value detainees were interrogated at Bagram and the Balad air base in Iraq.
The New York Times reports said the "black jail" was separate from the main detention centre, which houses about 700 inmates, including 30 foreigners.
That site will be closed next year when a new detention centre, which the US says has improved conditions, is opened nearby.
The former detainees interviewed by The New York Times
said they were held at the "black jail" site for 35 to 40 days.
All three were sent there after arriving at Bagram before being eventually transferred to the larger detention centre, which allows access to the Red Cross, the report said.
"They beat up other people in the black jail, but not me," Hamidullah was quoted as saying.
"But the problem was that they didn’t let me sleep. There was shouting noise so you couldn’t sleep."
The Washington Post newspaper on Friday reported allegations from two other former inmates that they were abused while at Bagram.
"That was the hardest time I have ever had in my life," Abdul Rashid, who said he is younger than 16, was quoted as saying.
"It was better to just kill me. But they would not kill me."
Rashid and another former inmate, Issa Mohammad, were reported as telling journalists that they had been beaten, photographed naked and deprived of sleep while they were held in solitary confinement.
It was not clear if they had been held at the so-called "black jail" or in the main detention centre.
Bryan Whitman, a US defence department spokesman, told The New York Times on Saturday that the military routinely sought to verify allegations of detainee abuse, and that it was looking into whether the two Afghan teenagers who spoke to The Post had been detained.
Without commenting specifically on the site at Bagram, Whitman said that the Pentagon’s policy required that all detainees in American custody in Afghanistan be treated humanely and according to US and international law.