Google says to update internal links from HTTP to HTTPS
If you are on an HTTP site and you decide to migrate the site to HTTPS, it is common practice to update all your internal links from HTTP URL format to HTTPS URL format. Google’s John Mueller was asked about this on Reddit and he replied “I would always try to fix the internal links, it keeps things cleaner and is under your own control. I doubt it will have any visible effect though. “
Generally, searching and replacing in your CMS and database from your HTTP version of your URLs to make them all HTTPS instead isn’t a huge benefit to your developers. If you have to do it manually, it’s more of a pain. But a search and replace can and probably should be done.
What’s harder is getting all your external links, links you don’t control, to get them to update it from HTTP to HTTPS. In that case, Google said in 2015, don’t bother.
Even for internal links John said “I doubt it will have any visible effect” because you have these redirects in place. But why not do it assuming you won’t break anything doing it?
Here’s the old tweet from Google’s Gary Illyes:
@pip_net If your redirects are implemented correctly, the benefit of doing so is so minimal that IMO it’s not worth it.
— Gary 鯨理／경리 Illyes (@method) July 7, 2015
So in short, if you can easily make the change with a search and replace, do it. Otherwise, don’t worry too much.
Discussion forum on Reddit.