Stay put or enter the US application process
While shane's argument has some merit, there is another important factor to take into consideration: time to degree, which is very much inequal here.

If you accept your professor's offer of a PhD position, you would in principle be able to start your PhD research as soon as this fall. If you were to decline the offer, you'd definitely have to wait a full year to start the PhD program, and then at least an extra semester or two, depending on the course requirements in your field. So there could be up to a two-year delay before you do substantial work in your PhD field, depending on whether or not the department will waive your master's coursework. This may or may not be enough to sway your decision.

Additionally, you should look at all of the "peripherals," including things like: 


how much teaching duty would be combined as part of the PhD position (this can be substantial, depending on your advisor's or department's teaching load!); 
what the benefits and support associated with the position are relative to the US programs you're considering; and 
what are the chances someone with your record will have to get into a comparable US program.