To add to the excellent answer by @can'tstopmenow, assuming that "working with him" does not include you moving countries or cities, I would make it clear in the letter that you are not looking to be paid, given supplies, help move (e.g. find a place to live, visas, etc.)
Not that this is the only criterion, but you have to answer the question from the professor's side, the "what's in it for me?" question. Not having a project to work on might be a good selling point, because you could offer to help with anything they need. So maybe your initial overture could start by asking how you can help them.
BTW, I think that the whole scheme is a great idea: you expand your network, and expose yourself to new ideas.