"He is at the presidential residence (in Niamey) with his wife and son and is doing well," the family member said, adding that he was allowed to make one phone call.
The source added that his doctor was able to see him and "bring him food."
On Thursday, the military regime which overthrew the democratically elected Bazoum on July 26 said they had foiled an attempt by him to escape their custody.
The escape plan, the regime's spokesman said, had involved Bazoum getting to a hideout on the outskirts of the capital Niamey before flying out on helicopters "belonging to a foreign power" towards Nigeria.
The regime added that "the main actors and some of the accomplices" were arrested.
A lawyer's collective representing Bazoum rejected the "fabricated accusations" and said Bazoum was being "held incommunicado."
French President Emmanuel Macron expressed Friday his "concern over the uncertain situation" and called "for his immediate release and that of his wife and son."
Since he was toppled by the military in July, Bazoum has refused to resign and has been held at his residence in the heart of the presidential palace along with his wife and son.
Last month, Bazoum's lawyers said he had filed a legal case with a court of the Economic Community of West African States (ECOWAS) against those who deposed him, and appealing for the restoration of constitutional order.
ECOWAS has warned that it could intervene militarily in Niger if diplomatic efforts to return Bazoum to power fail.
Former colonial power and ally to the deposed president in the fight against jihadist groups attacking Niger, France agreed with the military rulers demands to withdraw its 1,500 troops by Dec. 31.
France earlier pulled out troops from Mali and Burkina Faso, which have undergone coups in the past two years.
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