I think it would be okay to sit down with your direct supervisor maybe a month before your position is due to end. Come prepared with a list of new tasks you've taken on, trainings you've participated in, etc, and make your case for being an asset to the company. Tell them up front that you know they can't give you any information right now, but you want to make it clear how much you've enjoyed your time there and that you want to stay in some similar capacity regardless of what the employee on leave decides.
I don't think I would be annoyed by supervising someone who does a great job and clearly wants to be there. Yes, doing it now may come off as pushy and premature. And who knows, in the meantime, they may approach you with an offer, suggestion, or to ask how things are.
Regarding trainings, make friends with someone in HR or get on the list for a professional development calendar or something, then let your supervisor know you'd like to attend.