Hi Richard,
I’d like to confirm some information:
When setting up this account in Outlook client, will the following three steps be completed successfully?
If not, could you please capture a screen shot? You can click Use rich formatting in the forum, and then click the insert image button to post the screen shot.
About this issue: Outlook client keeps asking for password, there are some possible causes. This might be caused by that Office 365 Desktop Setup has not been run, or the Microsoft Online Services Sign In Assistant hasn’t been configured correctly. You can run Desktop Setup to resolve this issue: Set up your desktop for Office 365.
Therefore, I suggest you run Desktop Setup, and then set up this account in Outlook 2010 again to test this issue.
Thanks,
Young Yang