I am a foreign Biology student applying to USA phd programs.
I have contacted several PIs asking if they had an available position (because they work within my interests). I've got a lot of no responses, some "No funding/no position" and I've got some positive feedback. I have one particular university in which 3 PIs sent me back an "encourage to apply" mail.
Examples of these are:
You seem like a good fit (...)
I'll keep an eye out for your application
I might have some influence on comitee X
If you get accepted (I have no influence) ask to chat with me
Or just "I encourage you to apply".
From my point of view, the universities take into account that the PIs are willing to accept the candidate (they say the measure is "fit" not only grades). Then it would be more than wise to state clearly in my statement of purpose that I've spoken with Professor X about working there and received positive feedback. However, it might seem pedantic to state that "Professor X...". Just to make things clear, I'm not an outstanding, perfect grades, so I believe this could be a very important feature of my statement. Finally, it is also stated clearly on many programs that PIs have no influence on comitee decisions. I believe that there's some "legal hole" in which they don't want to give control to PIs but they find themselves needing PI's approval or something.
My question is: Should I include this information? How to do it politely?