Sending my research proposal for a professor or offering to work on their project?
Professors can get funding for projects if they write proposals for funding and the proposals are accepted. There are sources for funding for student projects in many places. I do not know about your specific situation, but I don't see why you wouldn't be able to approach the professor about your idea. If it's really good, the professor will help you find funding for it. If it is just good, the professor may have you find funding for it.

Full disclosure - I am a professor. :-)