If you receive error messages from Google when trying to open the Google Webmasters Tools/Search Console integration, ensure that the account used to login to Google is the same account associated with the website domain in Google Webmasters Tools/Search Console and that the domain specified is exactly the same registered in Google Webmasters Tools/Search Console.
The domain must match exactly including protocol and prefix and must be verified using the 'URL prefix' method, so if you have registered the domain http://www.mydomain.com and you are requesting for example http://mydomain.com Google won't allow the access. The same happens if you have registered the domain http://www.mydomain.com and your are requesting for example https://www.mydomain.com.
In these cases you will receive an error message: '(403) User does not have sufficient permission for site 'http://www.sitexyz.com/'
If needed, you can set the specific domain for Google Webmasters Tools/Search Console in the component configuration section as visible below:
If you are using the new Search Console to add a new property, ensure to use the 'URL prefix' method to register and verify your website. Indeed if you choose the 'Domain' property method based on DNS, the Search Console API won't work and you will continue to see the above error about missing permissions to access the requested site.