My approach would be:
- Book a 1 on 1 with your direct supervisor. If you have regularly scheduled 1 on 1's then you can just wait for your next one... but in a company without a regularly scheduled 1 on 1 culture, book a private meeting and mention that you want to checkup on your performance as an Intern.
- At the meeting - ask for feedback, mention your interest in pursuing a paid permanent position, and ask about the timeline.
- Do not - try to impose a fixed deadline ("I have to be a full time employee next month"), but do try to get sense for how the process will go, and when you should check in on the next step.
- Expect that unless your boss is already ahead of you, the transition could be as long as the originally cited 6 months total (so... if you're 3 months in, 3 months from now...). Employers may not necessarily expect to turn around a hiring action fast and the infrastructure of your company can make hiring a very slow experience that is beyond your direct supervisor's control.
My wording would be something along the lines of:
I'm really enjoying my internship, and the team. < insert some things you find particularly great - the interesting work, the smart or supportive coworkers, the relevance of the project, whatever it is you honestly like >.
I hope you and the team feel the same, or if you've got anything I can improve - please let me know. (pause for feedback).
(Unless the feedback is abysmal) - Thanks, that's really helpful. I wanted to ask - what's way to move forward with potentially joining the team full time? What does that process look like? How can we move forward? Do you need me to start doing anything to make that happen? (insert some info from you boss) - end with "OK, when is a good time for me to check back with you on this?"
There's a few edge cases here:
Case 1 - they don't love you as much as you love them... i.e., the feedback about your performance is surprisingly negative and you get the feeling you won't be offered a position.
Wow! I'm so sorry that (whatever is so bad) has been such a problem. If I'd known, I'd have corrected it earlier. Do I still have time to turn this around? Now that I know the problem, I'd like to show you what I can do with that feedback... mind if I check in with you again in a month? Does this hurt my chances of getting a permanent position at the end of my internship?
Yes, ask point blank if a performance problem has made you a no-go already - you deserve clear feedback, and it's easy for bosses to try to be vague and misunderstandings are common here. Keep asking questions until you know the severity of problems and the potential impact on your employment.
Case 2 - the feedback on your work is fine (or better) but the process for moving forward is inscrutable, or sounds unlikely.
Wow, that hiring process seems really complicated. When I started the internship, my impression was that a permanent position would be available within 3-6 months of my starting at the company. Has that changed? I had expected the conversion to full time to be fairly standardized and predictable. If that's not the case... should I be looking for a position elsewhere when my internship is over? I really like it here, but I need a full time, paying job.
That's a very soft way of saying "look, get it together - either hire me, or I'll stop making this (unpaid) position a priority". Not the thing to do if your feedback was bad, which is why you pull this trick ONLY when you are sure that (a) they love you, and (b) there is no clear path towards hiring.
NOTE - I wouldn't go making threats just because they plan to covert you at 6 months. The commitment to unpaid work was "3-6 months" - and there are tons of reasons why they can't/won't convert you on the early side of that estimate. But they should be able to be clear about it. An example of when not to fight it:
"We love you, the paperwork for a job offer is already underway, but I need to get budgetary approval to make you an offer. Due to the nature of our finances, I would MUCH rather ask for approval for the salary offer at the start of NEXT quarter rather than the end of this quarter... but that means I won't be able make you an offer until month 5 of your internship"
In normal person speak, this means:
"We blew the salary budget for this quarter. I might be able to squeeze your offer in, but you'd end up with a really lame starting salary. I would rather pay you what you're worth, but that means I need to be first in line when the next quarter's budget opens up."
Unless this was financially unacceptable to you, I would go with it. Having a strong salary helps your career in the long run, and it's worth a few months of delay.