An Azure relational database service.
I solved my own problem. First I needed to step away and think clearly. The answer is very simple. In order to solve this you will need to create the same SQL login in both Azure and Premise. From SSMS login with this account using the appropriate permissions assigned. You should be able to do this for Azure and Premise environment (if done correctly!). Once done you will be able to create your external data source. This has nothing to do with Linked Servers!