Disney is reportedly playing hardball with Apple over its TV service

Disney is putting pressure on Apple to include more of its channels in its much-rumored streaming TV service, according to The Street's Ronald Grover.

These would include spinoff channels of popular networks such as ESPN and the Disney Channel. Apple, however, wants to take fewer channels so that it can offer its service at a lower price, reports The Street, who claims to have spoken with people familiar with the matter.

The negotiations with Disney are especially important to Apple because channels owned by Disney, such as ESPN and ABC Family, are important to the cord-cutting demographic, The Street's source said. Disney CEO Bob Iger is on Apple's board of directors, which makes the battle particularly interesting.

Cord cutters are people who choose to opt away from traditional cable in favor of Internet-based streaming services such as Netflix or Hulu.

Apple is widely expected to announce a streaming TV service in June, but the service itself is expected to launch in the fall. The report follows others from The Wall Street Journal and The New York Times that said Apple is preparing to release its own TV service that could include about 25 channels.

It's clear, however, that Apple is looking to move into the streaming entertainment space. The company just launched HBO's new standalone streaming service, HBO Now, exclusively on the Apple TV, on Tuesday.

If Apple does decide to announce a TV service in June, we'll likely hear about it at the company's annual Worldwide Developer Conference.