Is it inappropriate to ask my supervisor (at a research intern) to review a cover letter for a PhD application? I recently finished my masters degree and I'm currently interning at the same university doing a research project. I'm planning to submit applications for PhD programs at other institutions and I've just finished my first application. The application includes a cover letter of 1000-1500 words. I've sent the cover letter to family members for feedback before I send it off who have given me great feedback regarding the style and language. Unfortunately, no one in my family is in academia and I would really like the thoughts of an academic since they're the target audience and they will understand the technical content of my cover letter. I was thinking of asking my current supervisor or perhaps my old supervisor from my masters program to give me feedback on the cover letter however I'm unsure if this is unprofessional or if I'm asking too much. 

My question is whether this is an unreasonable request?
It is, of course, a reasonable request (unless you believe your supervisor is unreasonable). 

Furthermore, the supervisor of your master's degree ought to be among the first to write your LoR. There is nothing wrong with requesting him/her to review your cover letter.