I've recently been in the position where my MSc thesis supervisor informed me about a possible PhD topic and suggested to apply. From this experience I would say that having an open discussion is best - of course depending on your relationship with the supervisor. You should present your case for wanting to pursue further study (this also helps to decide for yourself) possibly first by e-mail, however nothing beats a discussion in person. Academics are generally quite busy, so writing a long, serious e-mail is not the best approach. Adding a familiar face to your conversation will work better!
Are you aware of PhD opportunities at the university/with your supervisor? Can you address a specific project of his/of the department that you are interested in?
The great deciders:
Ultimately, a PhD is dependent on available funding. If your potential supervisor or you can secure funding for the proposed project (or best case - funding is already available), then you can apply for the position without a doubt. Of course, when a position has been funded usually the department will advertise to the world at large at which point other candidates will also be considered.
In any case - good luck!