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Every tool that I know of that makes normal maps, does so by first starting from a bump map, and a user-specified depth. In other words, the normal map is completely generated from a bump map and a specified desired depth, which means that the bump map is more general, while the normal map is baked permanently at that depth.

Zbrush and Mudbox, in my understanding of it, don't do it like that.
They calculate the difference of the sculpted detail on the high res mesh from the unsculpted detail of the low res mesh and create a normal map of it based on the mesh's UVs, not from a grayscale image.

I couldn't claim to know mathematically how it works, but I use ZBrush and it's a fascinating thing to watch, as it generates the normal map - it literally examines every polygon and compares it to its high res counterpart.

Now I suppose there are cases where grayscale bump maps are used to generate normal maps for .dds files, but that's really a hack - most normal maps these days are generated directly from the geometry itself.