I'm using Office Pro Plus 2016 on Windows 10 Pro 64-bit and having an annoying problem.

I have turned off Windows Update and Office Updates. This is working fine, I think. But I'm getting this annoying notification in Office:

UPDATES AVAILABLE Updates for Office are ready to be installed, but first we need to close some apps.

Coming accross to this notification every time I launch an Office product is starting to get to me. Isn't there a way to get rid of this?

Turned Windows Update off using the methods described in How to turn off Windows Update in Windows 10.

You are using the click-to-run version of Office 2016. Thus, the Office updates are not controlled by Windows Update.

When automatic updates are enabled for Office clcik-to-run version, in most cases updates are applied automatically in the background without any user input. However, updates can't be applied if an Office program is open. If an Office program is open, other attempts are made to apply the updates at a later time. If, after several days, updates haven't been applied, only then will users see a notification that an update to Office is available.

To disable the update notification, we can try the following registry key settings:

HKEY_LOCAL_MACHINE\software\microsoft\office\16.0\common\officeupdate

You can also create the hideupdatenotifications value with group policy setting to disable it:
HKEY_LOCAL_MACHINE\software\policies\microsoft\office\16.0\common\officeupdate