Google Inc. is close to launching a cloud-storage service that would rival one of Silicon Valley's hottest start-ups, cloud-storage provider Dropbox Inc., according to people familiar with the matter.

Like Dropbox, Google's storage service, called Drive, is a response to the growth of Internet-connected mobile devices like smartphones and tablets and the rise of "cloud computing," or storing files online so that they can be retrieved from multiple devices, these people said.

Drive...