Answered by Brittany, Hiring Expert at ManpowerGroup, on Thursday, May 25, 2017
If you are fulfilling your current internship until your agreed-upon end date, no, you do not need to tell your boss. However, it is okay to mention it and it's always good to ask if they are willing to write you a letter of recomendation and/or endorse you on LinkedIn. This will come in handy down for future positions.
Best of luck in your new internship!
Bri, former recruiter