"One of our top priorities is to bring iTunes Radio obviously here in the U.K. but everywhere in the world," Cue said last week in an interview from London. "We certainly want to be in more than 100 countries."
Apple launched its music streaming service late last month as part of an update to its iOS 7 operating system. Though a late entrant into the Internet radio service, in just its first week, it had 11 million users.
"We were very pleased, very pleased with the initial results," said Cue, senior vice president of Internet Software and Services.
In comparison, Pandora, the biggest online radio network, has 72 million listeners, while Spotify has 24 million. Both Spotify and Pandora are also available as an app on iPhones, while all new iPhone models come with iTunes Radio and some older models can update their systems to have iTunes Radio.
Cue says there's enough room for everyone, but adds: "We want to be the best." Its service has gotten some mixed reviews, but Cue believes it stands out because Apple has been able to create a better match of songs that listeners like.
"That's a lot of the feedback that I've seen both that people have written about and certainly the emails," he said. "It's the quality of the stations," he said. "The question and what the ability that we have that I felt was unique ... that we could have a radio station that played songs that you would really like."
Cue spoke as iTunes was wrapping up its annual festival in London, which saw Justin Timberlake, Lady Gaga, Katy Perry, Elton John and a host of others perform for the entire month of September at the Roundhouse Theatre. Fans worldwide could see the performances streamed live on an app designed for the festival and that were later available for purchase.
Cue said he'd like to see the festival expand its reach but keep its intimate setting at the Roundhouse, which holds about 3,000 people.
"We've had a lot of requests from places to take it bigger, but this I really believe is perfect," he said. "If you wanna see some of these stars in ... bigger arenas, there are definitely plenty of opportunities to do that."