Karzai was consulting with various government organisations to resolve the matter as soon as possible, the senior government official said on condition of anonymity.
"The president is personally working to resolve it peacefully. There is a way out of it," he said. "I believe it'll take one or two days."
Rahman was arrested under Islamic Sharia law about two weeks ago after his parents went to the authorities, reportedly following a family dispute.
Sharia law, on which the Afghan constitution is partly based, rules that a Muslim who converts from Islam should be put to death.
I'm sure this is a comfort to all the families whose sons and daughters and brothers and sisters have died to protect Afghanistan's right to execute Christians under sharia.