The same behavior is seen if the URL entered ends with /, the url towards the API gets transformed to /v1/job_templates/ instead of /api/v1/job_templates/.
Filip Reimer you are correct about the API transforming the urls to end with a /. In Jenkins, when you add a / to the end of the URL Jenkins ends up making calls to Tower like https://<server>//api/v2/ping (note the double slash at the start of the path in the URL. This is why the test fails with the message "The item does not exist". I just released an 0.8.3 version of the plugin which will string the ending / of the URL so the plugin should now work with a url of https://towerserver.something.com/ or https://towerserver.something.com. It normally takes a couple hours for the new version of the plugin to propagate. If you get a chance can you test this change and let me know if its now working either way for you?
Alexandr Suprunevich can you test the 0.8.3 version of the plugin and let me know if this change now remediates the issue you were having?