Close to follow, according to sources, will be a deal with Warner Music Group, the smallest of the "Big Three" music labels.
That would leave Sony Music Entertainment, the last of the Big Three, Sony / ATV, the music publishing company jointly run with the estate of the late pop legend Michael Jackson, to be signed before Apple could launch a web radio service similar to Pandora's. However, negotiations with those groups are not as far along in terms of reaching a deal, according to sources.
The rumored service, which has been dubbed iRadio by the media, will come with more on-demand features than Pandora, but will still be seen as a direct rival. While financial terms are still unclear, the sources said that despite its power, Apple will not receive the steep discounts it had been seeking, vs. Pandora's payments to the labels.
Last month, the New York Post reported that Apple had offered to pay 6 cents per 100 songs streamed, which would be about half of what Pandora pays. One of the sources said that instead, Apple will pay rates just about equivalent to Pandora's.
As of September 2012, the iTunes Store has 400 million active users, and in February, Apple reported that 25 billion songs had been purchased and downloaded from iTunes. Pandora said this week that it now has 200 million users.
Given Apple's huge market share in music -- since 2008, iTunes has been America's top music store -- as well as its user base, iRadio is seen as a huge threat to Pandora. Earlier on Thursday, Morgan Stanley analyst Katy Huberty wrote in a research note to investors that she expects Apple to unveil a streaming music service at WWDC in June.
Update: Apple reached a deal with Universal in early May.