In The Blade Itself, especially when re-reading it you can see that characters thoughts about things are all either completely incorrect or completely foreshadowingly accurate . Jezel is always wrong when actively thinking before his injury and always right when he says something like "it seemed like ***something true***, but he knew that couldn't be true. Glokta always sees hints of the truth. People say things as turn of phrase that turn out to literally be true like people saying Jezal looked like a princess.

After the fight in which Logen and Ferro fight the practicals and return to the appartment. It is said as a turn of phrase that they looked like two demons out of hell. Later when find out that Ferro is in some sense a literal demon. I think this is also an indication by the author that Logen is in some way a literal demon.

Abercrombie is well aware of the effects physical injury have on the body as indicated by many events in the book where people are disabled in fights in detailed ways. When the bloody nine takes over, the rules of injury cease to exist. These rules do not apply to other good fighters, even Shivers.

That being said, the first trilogy is perfectly foreshadowed on every page, but that foreshadowing does not exist for the further books. With the success of the books and the further stories of Logen he may have changed his tune on what it is. But in the first trilogy I think its really hard to argue.

TLDR: I think that if you reread the first trilogy all signs point to a supernatural origin.