I'm having the exact same issue. About a week ago two different google accounts started requesting my password but when I entered the correct password I would get the same message, 'Failed to authenticate with Google'. However I could still send and receive emails from that account on my Macbook, iPhone, and iPad even though all three devices said the same thing. I can't find a solution to fix this very annoying issue. I even set up 2 factor authentication thinking that would solve the issue, but to no avail.
In the past I have tried turning on SSL on both the pop and Imap accounts and here is what i get. 5.7.1: Client host rejected: Access denied. Cannot send mail. The SMTP server does not recognize any of the authentication methods supported by Outlook. My gmail account says there is a Authentication error! Help I am unabl to get into my email! I have important - Answered by a verified Email technician. My gmail account says there is a Authentication error! Help I am unabl to get into my email! Unfortunately unable to update or download any application using new gmail act.