Verizon FiOs is moving closer to unraveling the cable bundle with new skinny packages of channels that can be designed based on a customer's interests, a significant departure from the traditional model of selling consumers hundreds of channels at once.

FiOs, which serves 5 million customers nationwide, will begin Sunday to offer its new Custom TV service, which includes basic channels and seven small add-on packages grouped by genres including sports, kids, news, entertainment, pop culture and lifestyle.

In the basic plan, which will cost $54 per month, customers will get 35 basic channels including all local broadcast networks, CNN, HGTV, AMC and the Food Network. The package includes two genre packs. Consumers can add additional genre packs for $10 each.

But sports fans won't be able to watch ESPN or ESPN2 as part of the new bundles, which will turn off many cable customers who hang on to their subscriptions largely to watch sports events that aren't easily available online.

“Media reports about Verizon’s new contemplated bundles describe packages that would not be authorized by our existing agreements. Among other issues, our contracts clearly provide that neither ESPN nor ESPN2 may be distributed in a separate sports package," ESPN said in a statement late Friday.

Verizon FiOs President Tami Erwin told CNBC in an interview that the decision to shake up their offers was in response to new consumer demands. Subscriptions to cable and telecom television services have steadily declined in recent years. Streaming services such as Netflix, Amazon and HBO have become fixtures in American homes. Four out of ten homes subscribe to Netflix.

The decision also shows an acknowledgement by cable companies that as video migrates more to the Internet, the focus of their businesses will increasingly be on broadband, rather than cable.

"Customers want choice and increasingly customers have choice on video, and we've said very clearly we expect to be the preeminent broadband provider in the market, but we want to give customers choice on how they acquire and how they buy video," Erwin said.

She said some current FiOs customers may migrate to the cheaper plans but she said she also believes the new options will draw new customers.  "I think about millennials and how millennials are viewing video today for example," Erwin told CNBC.

It may be hard, though, to attract young consumers who have never paid for cable TV. With Internet included, the Verizon package will add up to well over $100 a month for consumers who today only pay for broadband plus a couple subscription streaming services.

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