The deal could be officially announced as early as Monday, according to articles in Fortune Magazine and The Wall Street Journal, both based on anonymous sources.
The purchase price is about $200 million above Quidsi's current valuation, Fortune reported.
Quidsi's management will stay on and join Amazon.com, according to the Journal.
Diapers.com sells a wide variety of baby products, including toys and cribs, while Soap.com focuses on household items and personal care products.
Quidsi launched Diapers.com in 2005, while Soap.com made its debut in July of this year. Quidsi's strategy has been to offer a very wide variety of products, coupled with low cost and very fast delivery and sharp customer service.
Amazon.com bought online shoe seller Zappos.com last year for $1.1 billion.
Neither Amazon.com nor Quidsi responded to requests for comment.