The Journal said Google is likely to announce the service at its annual Google I/O developers conference in San Francisco, adding that the service will operate in a testing mode initially and not be available to the public at large.
In April, Amazon unveiled a service that allows customers to store about 1,000 songs on its web servers for free instead of their own hard drives and play them over an Internet connection directly from web browsers.
Google could not immediately be reached by Reuters for comment.
The Journal said Google has not sought licenses from the four major recorded-music companies and the new feature is likely to include a system that functions much like a remote hard drive.
Users could listen to songs they have uploaded to the service in a "streaming" mode, but will not be able to download the files, the daily reported.
Earlier in the day, Google-owned YouTube said it will add more than 3,000 mainstream movies for users to rent from Monday.