If you are on an HTTP site and you decide to migrate the site to HTTPS, it is common practice to update all of your internal links from the HTTP URL format to the HTTPS URL format. Google’s John Mueller was asked about this on Reddit and he said “I’d always try to fix internal links, it just makes things cleaner, and is under your own control. I doubt it would have any visible effect though.”
Generally, doing a find and replace in your CMS and database from your HTTP version of your URLs to make them all HTTPS instead is not a huge lift for your developers. If you have to do that manually, that is more of a pain. But a find and replace can and probably should be done.
What is a bigger pain is getting all your external links, links you do not 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 would have any visible effect though” because you have those redirects in place. But why not do it assuming you won’t break anything while doing it?
Here is the old tweet from Gary Illyes of Google:
@pip_net if your redirects are properly implemented, the benefit from doing that is so minimal that IMO it’s not worth it.
— Gary 鯨理／경리 Illyes (@methode) July 7, 2015
So in short, if you can easily make the switch with a find and replace, do it. Otherwise, don’t worry too much about it.
Forum discussion at Reddit.
