iCloud sign in window keeps popping up despite being signed in

It may help to try restarting the Mac in safe mode, then try removing the iCloud account while in safe mode. Use safe mode to isolate issues with your Mac

If you're unable to remove it while in safe mode, try:

1. Go to Settings on iCloud.com, then look at the My Devices section.

   If you don’t see the My Devices section, your account just has access to iCloud web-only features. To gain access to other iCloud features, set up iCloud on your iOS device or Mac.

2. In the list of devices, click the one you want to remove.
3. Click the Delete button next to the device. If you don’t see the Delete button, click Remove Device (below the device’s serial or IMEI number).

Other people had luck by going into imessage, calendar, itunes, app store, and facetime and sign out of their account for each one.

If that still doesn't work, contact iCloud Support

Its not work either i got this "your icloud account couldn't be removed at this time" when press sign out

Open system preferences > click on iCloud you will see the box for keychain is unchecked, so check mark the box a small pop - up will come on the screen after a while to enter iCloud verification code so enter it .

Make sure your trusted mobile is opened , after a while six digit code will come through sms on the mobile .Enter the six digit code on the next pop up .

You will see iCloud Keychain is setting up and a small circle is rotating the box for keychain is configured with iCloud server .

Try this if possible post a screen shot of it ( hide your mail address ) .

1. Go to System Preferences > iCloud
2. Sign out then sign back in