Hi Matthew,
The issue is a little bit strange. Please share the screenshot of the pop up window which asks you for the password. To upload pictures, please click Use rich formatting when replying and then click the Insert/Edit Media button.
And I suggest you remove the current identity or profile and re-install the Outlook for Mac to see if the issue persists.
For Outlook for Mac 2011: Rebuild the Outlook for Mac 2011 database to resolve problems
For Outlook for Mac for Office 365:
a. Quite the Outlook for Mac program by clicking Outlook on the top menu bar > Quit Outlook.
b. Click Go> Applications.
c. Right click Microsoft Outlook> Show Package Contents > Contents> SharedSupport> Outlook Profile Manager.
d. Delete the existing profile and click the plus icon to create a new one.
e. Reopen Outlook for Mac to see whether it will connect to O365.
Thanks,
Henry Huang