Is it possible to download the non-DRM version of the game with the Steam client?
Define what you understand by "non-DRM'. If a game is inherently DRM-free (see here), you can download the DRM-free version from Steam; it will be put into your
steamapps/common dir - you can copy/move it out of there if you wish (note the possible caveats).
If the game is DRM-protected and the license allows for DRM-free use of the game or you don't care about fine legal aspects of the situation and just want to play the game you already bought (again, caveats), you may (with very high likelihood) succeed by using a cracked
steam_api.dll or its derivatives/substitutes. I won't provide the exact links for the obvious reasons (a quick Google search for e.g. "how to crack steam MPGH" or "steam emulator" can lead your way), but the generic
steam_api crack allows to use many (if not most) games in a DRM-free way. Note that you will have the exact Steam version of the game (updates, virus-free etc.), but you won't have access to Steam-related features (friends, achievements etc.); it's arguable whether this can be called "DRM-free version", but it most certainly will be downloaded from Steam.
Alternatively, just use your Steam on WINE. (caveats, yet again!)
NOTE: I don't condone piracy (copyright abuse/software theft) in any way. I, however, argue that a user has the right to break licensing terms (EULA) of the software he can legally use (bought/paid for or received free of charge) if they are exceedingly prohibitive (because that is the usual court ruling in such cases, both in EU and some US states) - the worst legal implication here is that the license may get revoked by the issuer (in this case, you may legally get banned from Steam for circumventing its protection). I assume that's a call everyone should consider by themselves.