notification "New accounts added" keep coming after I delete those accounts from the list

Are you using iCloud with your emails? If so, are there other devices (Macs, phones, tablets) where those old email accounts might still be active? Even if you no longer have those older devices, if they are still listed as iCloud devices (you can remove devices that you no longer use) the email account might be trying to reestablish itself because of iCloud. You might log into your iCloud account (icloud.com) directly and see what devices are there and what email accounts are shown there.

ingenieurgeneral wrote:

hi leroydouglas, well yes Safari is my default browser. In Mail app I have only the accounts I need, not the ones that keep coming back (I just checked again, nothing to delete under Mail>Settings>Accounts.

Try a SafeBoot  to clear system cache files—How to use safe mode on your Mac - Apple Support

Takes noticeable longer to get to the login screen, does a 5 minute disk repair before it fully boots up, and certain system caches get cleared and rebuilt, including dynamic loader cache, etc.

Login and test. Reboot as normal and test. Caches get rebuilt automatically.

ly.

ingenieurgeneral wrote:

Hi, I am on a MacBook Pro (14-inch, 2021) with macOS Monterey (12.2.1). Every day, I get the notification "new accounts added", I click on "continue", it opens the "Internet Accounts" screen in Settings. Surprise: I see some old email accounts that I haven't used since many years. I delete those accounts by clicking on the minus "-" sign on the bottom left.

I don't know how Apple "finds" those old email accounts, it's quite annoying in terms of privacy, and what is even more annoying is that this notifications of "new accounts added" keeps coming back every day (and nothing weird has been installed on the computer, it's brand new!)

https://discussions.apple.com/content/attachment/9181fdfb-8b4b-4a00-ac18-ed54cb86d1df

Do you likewise delete these account in your Mail>Preferences...?

it may seem like a longshot—however verify you have Safari set as your default browser and test again your removal.

Mail User Guide for Mac - Apple Support