I can't think of any podcast app that supports that. They are meant to consume, present and harvest online feeds as their source.
Podcast addict will certainly import a directory.
Yeah I was going to bring up Podcast Addict as well. Dunno why you'd use anything else.
Nice, thank you! For me there are several no-gos, such as advertisements and it not being FOSS but for the majority of people it looks like the perfect choice.
Oh I paid for the premium version years ago. Well worth it.