How long does it take to get a written offer after a verbal offer? (Internship)

Apologies if this is a redundant question, or if I'm worried over nothing.

I had an interview with a tech company last week, and it went great. Towards the end, the interviewer asked if I was looking anywhere else and I told him that I had received an interview for an internship in another department of the same company. He said that he definitely wanted to hire me and was going to fill out the paperwork for an offer, but to go for that other interview and make my own choice. The other interview was today, and I haven't heard back about the first position (which is the one I think I'll accept). Should I be worried? How long should I wait to reach out?

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