According to a report in the New York Post, Apple and are in advanced talks and a deal is expected to be announced shortly.
Apple launched its service, selling music online, at the end of April. It offers songs for 99 cents and no subscription fee, with a catalogue of more than 200,000 songs including Eminem, Bob Dylan and U2.
The service, , offers personal rights use, including burning songs on to an unlimited number of CDs for personal use.
The service, which saw 1m songs downloaded in the first week, is currently only available to Mac users, but is expected to be available to Windows users later this year.
The deal would benefit both companies, giving Apple an important foothold outside of the Macintosh world and allowing Amazon, which recently moved into kitchen and houseware, to expand its ever-growing list of products into the downloadable music arena.
At Amazon's annual general meeting last week in Seattle, Amazon chief executive officer Jeff Bezos told that the company has been looking "for years" at breaking into this market.
Speaking at the launch of iTunes, Steve Jobs, CEO of Apple, said: "Consumers don't like to be treated like criminals and artists don't want their valuable work stolen. The iTunes Music Store offers a groundbreaking solution for both."
Users can listen to a free 30-second preview of any song in the store before buying.
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