The New York Times reports "three people briefed on the plans" as saying that Apple is in the early stages of negotiations with the major labels over the web radio service, although the "full scope of its plans were not clear".
Apple already dominates the market for digital music downloads via its iTunes Store, but personalised radio services from the likes of Spotify, We7 and Pandora are growing in popularity.
According to The New York Times, the Apple radio service would most likely comes as a pre-installed app on iOS devices, such as the iPhone and iPad. It may also connect to a user's iTunes account to judge their music tastes.
Most internet radio services currently operate under strict licensing conditions, meaning there is a limit on the number times certain songs can be played in any given hour.
But the sources said that Apple wants direct licences with the record labels, giving it much more flexibility over use of the music.
To make money from the scheme, it is thought that Apple would sell advertising through its iAd platform, although it is unclear whether it would share some of this revenue with the labels, or just pay them a licence fee.
It is also unclear whether Apple would offer the service for free to iOS users, or charge them a fee for an advert-free version.
Steve Jobs, Apple's late chief executive, always resisted plans to launch a music subscription service, as he felt that people did not want to rent songs.
However, the growth of smartphones, tablets and devices that are almost always connected to the internet has meant that many people no longer want to store their own libraries of music, and would rather just tap into a tailored service.
Nokia recently rolled out a new radio service in the US that offers buyers of its Lumia smartphones up to 150 music playlists to stream for free on their handsets.
The New York Times notes that the Apple radio service "did not seem to be imminent" and negotiations with the major labels are notoriously difficult.
Virgin Media announced its own music streaming service in 2009, but ultimately dropped the plans after talks with the labels stalled, and then partnered with Spotify for an app on its TiVo platform.
The people briefed on the Apple plans said that it was unlikely that the custom radio feature would be announced alongside the expected launch of the new iPhone next Wednesday (September 12).
Apple declined to comment on the speculation, as did representatives of Sony Music, EMI, the Warner Music Group and Universal.