You can get at this information by querying a view named "_views" in Tableau Server's metadata store. You'll want the view_url column.
Details on how to access our metadata store can be found here:
How can i Access Meta data of data source? Like which dimensions & measures that data source have? and many more.
Different tack... but you mentioned API: it is perhaps also possible using rest api query workbooks for user - that has the additional benefit of knowing which workbooks a user has access to.
(hope this message does not come iwth 10 links... I had some trouble adding the link to the message)
Hope this helps,