The authorities entered the flat after being alerted by a caretaker, whose suspicions were aroused by the smell on the stairs, local media say.
They had lived together for five years or more, since the woman took the man in from a life on the streets.
When a fall two years ago largely confined her to the flat, he stayed on.
Forensics experts believe the man, Didier Delavigne, died first and the woman, Elisabeth Devidas, died shortly afterwards, the French newspaper Sud Ouest reports.
The woman was found on Monday afternoon with her arm wrapped around her companion in their bed in the flat, close to Bordeaux railway station.
People who knew them told the paper they had made an “odd couple of iconoclasts”, walking hand in hand in the street at the start of their relationship.
“They say he was homeless and you could smell the wine from 10 metres away,” Heri Rananaivoson, the caretaker, recalled of Didier Delavigne, when he first made his appearance six or seven years ago.
But he seems to have changed after he met Elisabeth Devidas, for whom he initially ran errands before moving in.
“I saw him on the mend,” said Heri Rananaivoson.
“He stopped smelling of alcohol.”
The last time he saw Didier Delavigne, at the beginning of the month, he was “looking well”.
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