Not invited for on-site interview after phone session I had the chance to talk to two professors last week regarding admission to the PhD program. After I was done, I was told that I would be invited for an on-site interview, if admitted. 

After not receiving any feedback from the school, I emailed them and found out that I was not invited to the interviews on the campus. Now they're just telling me to wait and I may be invited for another visit in the future.

This is a problem for me as I need to start looking for funds in another programs that I have been already admitted to (The one I am waiting on is my first choice).

What would be the best approach to handle this kind of situation?

Thanks.
Exactly what they told you to do: wait.  If you're at the point where you'll need to make an irreversible decision (like accepting an offer somewhere else), you can tell them that, and see if they react, but it doesn't seem to likely to me.  Otherwise, what's wrong with waiting?