The pair were arrested late Thursday as they were fixing explosives to themselves behind the governor's residence in Ghazni, provincial government spokesman Ismail Jahangir told AFP.
"They both were attempting to get into governors' compound and target the governor and high-ranking officials," Jahangir said.
It is rare that women carry out attacks in Afghanistan's insurgency, which is led by the Taliban movement that was in government between 1996 and 2001 and is said to have support from extremist circles based in Pakistan.
A suicide attack in May in southwestern Farah province was apparently carried out by a woman in an all-covering burqa.
Jahangir said the woman and the child could not speak either of Afghanistan's main languages, Dari and Pashtu, but spoke Pakistan's Urdu and Arabic.
The pair were presented to the media several hours after their arrest.
The deputy police chief of Ghazni, Ab