Converts born with a Jewish soul (but born to the "wrong" parents), or converts getting a Jewish soul upon conversion?
I've heard the idea that if somebody converts to Judaism then it means that they have felt Jewish, and the conversion is more of a confirmation. And it means that they have a Jewish soul, and were born to the wrong parents, in the sense of, given that there parents aren't Jewish, it wasn't clear that they had a Jewish soul. The conversion is a test that shows that they are Jewish, (so, i.e. their Jewish soul led them to that path).
But I recently heard another idea that upon conversion, God gives them a Jewish soul. I've never heard that one before.
Are there sources for the "born to the wrong parents" idea? (I'm sure there are, but which?).
And for the "given a Jewish soul upon conversion" idea that I just heard?