Drèze told The Hindu newspaper earlier this week that he had not written the book. He did not respond to a request for comment from BBC Tamil.
The library was built at a cost of about 3 billion rupees ($34 million; £26 million).
The controversy centres on disputed titles whose selection for the collection has been challenged, including a book that a named author said he did not write.
His statement raised concerns that his name may have been used without his permission.
It remains unclear whether artificial intelligence was used to produce any of the disputed books.
More than 160 writers, researchers, artists and publishers signed a joint statement calling for an independent inquiry, the release of procurement records and an audit of library collections across Tamil Nadu.
The group questioned how the disputed books could have passed a selection process involving expert committees, price negotiations and several levels of review.
The row has prompted writers, academics and publishers to ask how books were chosen for public libraries.
The case has raised wider questions about the growing use of AI in publishing, where tools can generate entire books quickly and cheaply.
Those tools can also produce false information and invented references.
The dispute has highlighted the difficulty libraries and publishers face in checking the authenticity of books as AI-generated content becomes more widespread.
The controversy also sparked a political row in Tamil Nadu.
Rajmohan accused the previous DMK government of ordering the disputed books and said the state would seek to recover public money if the investigation found wrongdoing.