I use Microsoft Outlook 2011 on my Mac, but sometimes popped up the password request window even if the password is saved in my keychain and Outlook can access it.
The answer is described in the following site.
Fix Outlook 2011 forgetting passwords | MacIssues
The procedure is below.
- Quit Outlook if it running
- Select Microsoft Outlook.app in Finder
- Open Info window by CommandKey with I
- Check Prevent App Nap
- Re-launch Outlook
- Input the mail password