Al-Qaeda in the Islamic Maghreb has claimed responsibility for the January 15 abduction of Dr Ken Elliott and his wife Jocelyn following an attack on the capital Ouagadougou that killed 28 people.
In an audio statement, it said it will release its female captive unconditionally.
"The primary motive behind their kidnapping was an attempt to (gain) release of our captives who sit behind bars and suffer the pain of imprisonment, as well as being deprived of their basic rights," the recording by the al-Mourabitoun branch said.
The group said it was releasing the woman under public pressure and in accordance with what it said was guidance from al-Qaeda leaders not to involve women in war.
The West Australian couple, aged in their 80s, moved to Burkina Faso in 1972 to set up a medical clinic in the town of Djibo in the country's north.
Dr Elliott performed 150 surgical operations a month in the Djibo hospital he designed and built, according to a 2013 Global Business Services newsletter.
He is the sole surgeon at the clinic and has spoken in the past about the significant need for modern medicine in the West Africa.
