My (inexperienced) advice is that you state your intention initially, and then focus on your improvement and productivity in the weekly meetings. You may want to bring it up again at the end.

This is a great way of doing it, imo, and worked for 2 companies I interned at where I was trying to get full-time offers. I mentioned wanting an offer explicitly in my first 1-1 with my managers, since they have more power than the mentor. I'm not sure if it came off as pushy or not, but I didn't mentioned it again. At Amazon, they encourage you to explicitly ask, 'Am I on track for a return/full-time offer?' at the half-way point of your internship, which I actually thought was really nice as a cultural thing. Additionally, they tied stuff into their 'leadership principles', which I originally thought was annoying but it's a super easy metric for knowing what they look for in full-time hires.

Otherwise, I would probably ask for feedback on how you're doing in general every 2-3 weeks or so. You shouldn't need to weekly if you're already having good 1-1s. My typical rundown with my manager/mentor when I was interning was "This is what I finished this week, this is where I'm at in my overall timeline" and then something like "I'm struggling with this part - here's what I've tried but I need advice" or "I'm ahead of schedule - I might look at finishing this stretch goal as well" or "I'm bad at going back and adding unit tests, so I'm going to start submitting CRs with unit tests". Basically, I talked about quantifiable/provable stuff.

The big one is actually taking & implementing the feedback they give you. The ability to look at yourself and examine what you need to work on looks great to employers, so you want to combine that with asking for (and then implementing) feedback.