Police said the victims had not been formally identified but were believed to Yvonne Walsh, 25, and her seven-month-old son Harrison Walsh.
The bodies were found at 10:30 BST on Sunday in Chells Grove, Billesley.
A 28-year-old man was later arrested on suspicion of both murders. He is currently in a secure unit.
West Midlands Police said mental health professionals carried out an assessment and deemed him unfit to be questioned at present.
A spokesman said the man was known to the victims and police were not looking for anyone else in connection with the deaths.
The Independent Police Complaints Commission has been informed, due to previous contact between police and the man in custody.
Post-mortem examinations will be carried out later.
Forensic tests were still being carried out at the house, which remains cordoned off, on Monday morning.
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