The rumors are apparently true. Google is testing a method that lets users log in without using a password. A user on Reddit who is part of the beta posted screenshots outlining the way it works.

“Once a user authorizes their mobile device, they’re able to input 
their account credentials on any computer and receive a notification 
on their smartphone. The device must have some type of screen lock 
security feature, as unlocking your phone is a prerequisite to approving 
or denying access to the account with this method. You’re still given 
the option to log in with your regular password if you so choose. 
Google’s email to participating users explains how you can deactivate 
a lost device, as well as add a new one in the event you upgrade 
your handset.”

Google is testing a way for users to log in without a password – The Verge