A couple claimed they paid 120,000 rupees (£1,330) to Anima Indwar, who worked at a shelter for unwed mothers run by the Missionaries of Charity.
The arrests on Tuesday and Wednesday followed the couple's complaint the charity worker had taken back the baby boy and kept their money.
The baby, born to one of the shelter's residents, is now in the state government's care.
Spokeswoman Sunita Kumar said the Missionaries of Charity was investigating, but "there was no question of selling any child as the Missionaries of Charity had stopped giving children for adoption three years ago".
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