You need separate tool-chains for both
Note, the nightly build page gives you 64bit builds, so there isn't an awful lot of need to build yourself unless your trying to add patches.
0.10 Final (hopefully out soon) will be the last available 32bit build from us, and after that there is no guarantees that 32bit will work, or even build. (Note, the beta page still has 32bit builds)
http://sourceforge.net/projects/mingw-w ... %20Builds/ mingw-w64-bin_x86_64-linux_*
http://sourceforge.net/projects/mingw-w ... %20Builds/ mingw-w32-bin_x86_64-linux_*
This assumes your linux VM is 64bit. If it's not, you need the i686 equivalents of those packages.
I haven't tested, nor have time to, the latest releases of MinGW, and they don't provide the exact toolchain version I use any more. Anyway, you can give this a shot and let us know either way. It really is as simple as extracting these tar files, and adding the "bin" folder to your PATH.