The warning - first introduced last June and then repeated on occasion - is popping up both for users whose accounts may have been compromised and for those whose accounts have not but are under attack.
According to Eleven Media Group, the warning has been received by their own chief editor, as well as the editor of The Voice Weekly Journal, Myanmar correspondents for the Associated Press and the Kyodo News Agency, some IT experts and other Gmail users.
Also, in the last month, the websites and Facebook pages of many of these media outlets have been under DDoS attack, hacked and defaced.
Users who have received the notice have been advised to change their password to something longer and more complex, to enable 2-step verification, to update and keep updated their OS, browser, document editors and plugins, and to be on the lookout for phishing pages impersonating Google.
It is still unknown what state(s) are behind the attacks.