It should not work.
But, if you are restoring an image, then why are you running a 32-bit OS? IIRC, Windows 7 will accept the same key for the 64-bit version, so just backup and go 64-bit.
Even if it doesn't work, once you get having the 3+GB RAM, you'll really want 64-bit anyway, unless you're stuck with an incompatible 32-bit vertical market application.