The move is part of a deal between the two countries to transfer all Afghan prisons back to authorities ahead of the withdrawal of the North Atlantic Treaty Organisation forces by the end of 2014.
Afghan President Hamid Karzai declared the planned handover of the Bagram prison as a victory for national sovereignty.
According to the BBC, more than 3,000 inmates are held in the prison, including about 50 foreigners not covered in the handover agreement signed in March.
The US military still wants to run a section of the jail and is not handing over some detainees, saying it has the right to hold insurgents caught on the battlefield, the report said.
The US is concerned that some top-level inmates could be released or tortured if they are handed over.
The Bagram prison has been described as 'Afghanistan's Guantanamo' for prisoner abuse and indefinite detention.
In April 2010, a BBC investigation uncovered allegations of prisoner abuse at a hidden facility at Bagram.
The US military, however, denied it was operating a secret jail.
In January 2012, Afghan investigators accused the US army of abusing detainees at the Bagram prison.
According to the report, the investigators said prisoners had reported being tortured, held without evidence and subjected to humiliating body searches.