I've never heard of asm.js before, but doing a bit of googling looks like it's sort of like an LLVM backend.
Theoretically, yes. However, that translation from CIL to LLVM IR isn't the easiest thing in the world. Right now I'm only supporting a subset of the instructions that roughly correspond to the things available on the GPU, and adding support for the entire C# language is an extreme task. There's a reason there's a huge team at MS dedicated to writing the JIT for .net.
But yes, it would be possible, it would just be a lot of work and probably wouldn't support much of C#.