Should I inform my manager early that my internship is ending next month?

Hi everyone, I need a bit of advice.

My internship is scheduled to end on 13 April, and today is 27 March, so there are around 2–3 weeks left. I’m not sure whether HR usually informs the manager about this in advance or not.

I was thinking of sending a short email to my manager as an early heads-up that my internship is about to conclude on 13 April. I am currently in my 8th semester, and my final exams are from 20–23 April, so after that I will be fully available for full-time work from the end of April.

Also, I am open to either:

a full-time opportunity after April, or

an extension of the internship for now.

Would it be a good idea to send this email early, or should I wait for HR to initiate the discussion? I don’t want it to look like I’m rushing, but at the same time I don’t want to miss the chance either.

Any suggestions from people who have been in a similar situation would really help 🙏