Apple Inc. is stitching together a network of Internet infrastructure capable of delivering large amounts of content to customers, giving the company more control over the distribution of its online offerings while laying the groundwork for more traffic if it decides to move deeper into television.

Apple's online delivery needs have grown in the last few years, driven by its iCloud service for storing users' data and rising sales of music, videos and games from iTunes and the App Store. But the iPhone maker is reported to...