Instagram offers different privacy settings for your "Close Friends" list. When you add someone to your Close Friends list, they won't be notified, but they'll be able to see that they are on your list if they visit your profile and look for the green badge with a white star in the top right corner.
However, the list itself is not visible to other users, and they won't be able to see the other people you've added to your Close Friends list. It is designed to be a private feature, allowing you to share certain stories with a selected group of people you trust.
Since social media platforms often update their features and privacy settings, it's a good idea to check Instagram's official documentation or the app's settings for the most up-to-date information on this specific feature. To do this, go to your Instagram profile, tap on the three horizontal lines (hamburger icon) in the top right corner, then go to "Close Friends" under "Settings" to review the current privacy settings for your Close Friends list.