Can I use the exact same research proposal for different scholarships? I wrote a research proposal (edited by my potential advisor) for an external scholarship application. This application is due at the end of the month. 

I just found out that the internal scholarship also requires a research proposal. Would it be ethical to use to exact same proposal (i.e., copy and paste) for the internal scholarship application which is due at the end of next month? Or would this be considered plagiarism? 

The department will have both of my scholarship applications. So they will be able to see they are the same.
No problem, often people in your situation do the same. Even professors try to improve failed grant proposals in order to submit it again in other calls. 

In your case, I think you must send the same project, since you have already delineated the work you will perform with your potential supervisor and you are just applying for funding. After the results, if both get approved, you will need to choose only one of the grants. 

Plagiarism is a completely different thing:


  the practice of taking someone else's work or ideas and passing them off as one's own.


In any case, I think you should talk to your supervisor before submitting.