I am an American citizen and want to attend a Singapore University for my further studies. For PR, I have to pass any national-level exam which I have not. Will I face any problems with my PR application in the future?

If you enroll in a recognized university in Singapore and get a degree or diploma, it can help you get a job in Singapore that requires an S Pass or Employment Pass. Once you obtain one of these passes, you become eligible to apply for Singapore PR, even if you could not pass any national-level exam in the past.

National level examination is relevant for applying for PR under student quota. Since you did not pass that you are not eligible to apply for PR under those schemes. However the fact that you have done some education is Singapore is in favor of you in your Singapore PR application under the PTS Scheme.