The Business Insider US technology blog first reported the potential deal, claiming that it is part of Apple's further move into the e-commerce market.
Thefancy.com bears the slogan "Discover amazing stuff, collect the things you love, unlock crazy good deals".
Users are able to compile 'wish lists' featuring photos of things that they "fancy", but also access web links to buy the items directly through the site.
The format has been likened to Pinterest, the popular social network that involves users 'pinning' images of their favourite items to an online board.
Pinterest is often used by brands for marketing purposes, but The Fancy has been developed with a much more integrated focus on e-commerce, involving the site taking a 10% cut of sales.
In a blurb on thefancy.com, the company describes itself as "part store, blog, magazine and wishlist".
"It's a place to discover great stuff, to curate a collection of things you love, to get updates on your favourite brands and stores and to share your discoveries," it adds.
"Use it to create a catalogue of your favourite things around the web and around the world. When you find something you love somewhere on the web, Fancy it."
The report noted that the 20-person New York startup, led by co-founder and chief executive Joe Einhorn, is "much farther along in linking its users to transactions" than rival Pinterest.
It also claimed that sales on The Fancy were "exploding".
Taken as an indicator that a deal is close, Business Insider said that Einhorn met with Apple chief executive Tim Cook at the annual Sun Valley conference earlier in the year, after which Cook - not known for a penchant for social networking - started using The Fancy personally.
The website said that no deal had been agreed and there was no indication as to what price The Fancy would command.
The social network recently raised $10m at a valuation of around $100m, so any acquisition would likely prove costly for Apple - although the world's most valuable technology firm would not struggle to find the cash.
The Fancy already has strong backing in the tech community - Twitter co-founder Jack Dorsey and Facebook co-founder Chris Hughes serve on its board, while Marc Andreessen and Ben Horowitz, the co-founders of venture capital firm Andreessen Horowitz, are among its investors.
Should the deal materialise, then Apple will no doubt hope for better success than it has with its previous social networking experiment. It is expected that Apple will soon ditch the Ping music network, after it failed to ignite the interest of users.