Is it possible for a PhD applicant to be rejected against the potential supervisor's will? I have been in contact with a potential supervisor in one of tier two US universities before applying and all the way to receiving my rejection letter a few days ago.

The individual department I applied for has seven faculty members including my potential supervisor and as I know he is not on admission committee.

I try to quote his responses to me in our email exchanges.

When I first contacted him in October to show my interest in applying for Fall 2017 semester he replied: "I am interested to have you as my PhD student. But, you should apply fast to be here by Spring Semester. I can get you on a research funding that I have for next semester on ***** Project."

I couldn't prepare my application materials on time and I was told: "It is too hard to get the I-20 in a timely manner that you can be here for Spring semester. So, I am changing my plan of the project to be able to support you for the next fall."

I updated my application for Fall 2017 semester.

Last time before receiving my decision he replied, "Your application has come to college and is under consideration, I supported your application strongly. We should see the result soon." to my email regarding application status.

After receiving my rejection letter I emailed him to let him know I was rejected and thanked him for his support during this period.

But I think he was as shocked as I was on hearing that I was rejected.
He replied: "Why, your application was rejected!!
Please let me know, I should be able to help you."
and "Everything were positive on your application from my last communication. I am thinking that  your application may not have gone from **** Department to college of Engineering and to Graduate school, I will check on that tomorrow morning and I will let you know."

Do you think is it possible a rejected applicant being accepted when their potential supervisor really wants them?
Yes, it's possible.

Practices vary, but in most US universities, graduate admissions are not decided by individual professors, but rather by a committee of faculty.  Their decisions often must also be approved by a graduate dean or another higher-level committee.

There are many possible reasons why an applicant could be rejected, even if they have the support of a potential supervisor:


The committee might disagree with the supervisor's assessment of your qualifications.
The professor might have a record of questionable judgment in recommending PhD students.
Many PhD programs include coursework and exam components before beginning to work on dissertation research with an advisor.  It could be that the committee isn't confident that you would pass those components.
The program, or the university, might have strict minimum GPA or GRE score requirements, that you might not have met.
The committee might not feel that the professor would be a good supervisor.  This could happen if he does not have a successful record with previous students, or if they feel he already has too many students to advise well.
The program might have a limited capacity for students, or a limited amount of funding.  Even if the committee agrees with the professor that you are well qualified, there might still be more qualified candidates than they can accept.  Maybe there are even too many candidates who have the support of particular professors. If so, some good candidates have to be rejected.  This is probably the most likely reason.
It could be, as the professor seems to think, that a mistake has been made, and his recommendation has been unintentionally overlooked.  If that is so, it is possible, though not certain, that the decision could be changed.