when to reach out to HR/interviewer

I've been networking/informally interviewing for a lending role at a MM firm. Last Wednesday I had an interview with the hiring manager, who mentioned that he'd be reaching out to HR to set up additional interviews. He said they'd likely reach out by the end of last week but that it could be delayed due to slow-moving bureaucracy. I still haven't heard back from anyone at the firm. When should I reach out again? What should I say in that email? Should it be to HR, or the hiring manager?