Does my supervisor want me to do my Ph.D. with him or elsewhere?

As Peter said in his comment, ask him.

However, having said that, and based on what you have said, it is very clear that he is not 'kicking you out' at all - on the contrary, he is demonstrating a very positive and constructive approach - by telling you that you have to pursue what is best for you.

Having said that, you need to consider a key question:

• What is/are your goal(s) after you complete your research?

The professor is, as you say not an expert in the field you are interested in, and what he is suggesting is that, as this is the case then the internship may lead to opportunities that are more in line for your plans.

He is also seemingly suggesting is to pursue the internship and then decide where and what you want to study for your PhD.

Ultimately, the decision is yours to make - he is helping you to help yourself in this regards.

But, a piece of advice, do not assume to know the intentions of anyone - take his advice on face value and in the spirit that it has been given.

One of the biggest temptations for many advisors is to hold on to a productive student or postdoc a little bit longer than is necessary just to reap the benefits of a productive worker. Against the personal gains of the advisor need to be set the goals of doing what is in the best interests of the student.

That means that a good advisor will try to make sure that the student can explore all of her options. I'm sure that he would be more than willing to keep you around, if he's mentioned co-supervision. But if you really want to go elsewhere, then he really doesn't want to be seen as trying to strong-arm you into staying. That could build feelings of resentfulness that could poison your future working relationship.

Now, on the other hand, if you complete this internship, and then decide that the project with this advisor is still the route you want, then you have done your due diligence—which is what a good advisor would want you to do in any case!

Your supervisor just wants to make sure you make the right decision. Therefore, it might seem wiser to stay where you are now with him (plus another co-supervisor) if you know you get on well than go to another (even more prestigious) university where you might not find people you really want to work with (although they are in the field you are interested in). On the other hand, internship gives you the opportunity to meet faculty of different university and see how you can get on which is just as important as academic qualities of your new supervisor. Your current supervisor sees that as a great option for you to see where you want to do your Ph.D.

I think this is not such an uncommon thing in academia. Although it would help your current supervisor if you stayed (you can teach some classes, more research gets done in his group etc.), he can also see you as a possible future collaborator. From this point of view, it is important for him that you get the best from your Ph.D. even if it means leaving your current university.

I myself was in a similar situation a while back. Although my supervisor would like me to stay for Ph.D. he knew that there are universities and research groups where I can learn more and he did all he could to help me choose the one that would suit me best.