I was having the same problem for quite a long time now, until today, when I finally decided to do something about it. Most of the solutions/answers I came across recommended installing external software, or other complicated things. Then I decided to look under the hood itself.
Go to Chrome Settings, scroll down to Show Advanced Settings, scroll down again to the section Passwords and Forms. There should be an option - "Offer to save your web passwords". Ensure that the box next to it is checked.
For some reason, this box was unchecked in my settings. I'm not quite sure how it happened, but once I checked this box, for every password update I made, I would see the pop-up box "Do you want to save this password...."
Hope this helps.