As far as has ever been discovered - no, this won't work. You can think of a fan as just a prop enclosed in a housing (jet engine). A prop loses effectiveness at the speed of sound because the air passing the prop gets a high pressure shock wave that effectively makes the prop not prop-shaped, so it can't move air. Picture it as using a hammer on glass instead of pushing on glass with your hand, one works way better than the other for generating thrust. The only way (so far) to have an effective prop is to have the prop tips move slower than the speed of sound. Engine designers worked around this by slowing the air down around the prop. To do this, they moved it inside a tube. They slowed the air down so that the prop (fan) can travel more slowly, and then they heat the air behind the prop to gain excess pressure and thrust. And this is exactly what a mach 1+ jet engine does. The opening at the front of the engine forces air in, and the design of the inlet slows the air down so it is subsonic, along with a corresponding increase in pressure. Some fans (props) which are now effective because they spin at subsonic speeds compress it more, so more air can enter the engine. Then they burn fuel to heat up the air, increasing the pressure. After that they have a few more fans that run in reverse to drive the fans at the front of the engine, and finally exhaust this hotter, bigger, more high pressure air out the back of the engine to produce thrust.

So hopefully you see how your question is an interesting one, and one that has already been sorta done. Turbofans and turboprops are really quite similar, but at mach speeds only the turbofans have the right environment to be able to work efficiently. Your idea would have the prop in a supersonic air stream, which would make it effectively useless.