Valve worked on something like this a few years ago, while porting their engine to Linux: https://github.com/ValveSoftware/ToGL
. There was also this effort
to translate DirectX11 to Vulkan.
In a technical sense, it's totally possible to port DirectX to Linux, by either translating calls to a Linux supported API like OpenGL or Vulkan, or by implementing a driver. I can't speak about the legal aspects, however this is probably why there's no official DirectX driver on Linux.