I have completed several rounds of interviews for an assistant professorship position with a university. In June, the department chair verbally informed me that they wanted to hire me. At their request, I presented my preferences for the start-up package (e.g., lab space, start-up funding, number of postdocs and PhD students) and my proposed start date, which they were okay with. Following this, I presented my research funding plans. Subsequently, I received an email stating that my case had been forwarded to the dean for approval. I was also verbally assured that the dean had been involved throughout my interview process and related discussions.
Given the positive feedback from the department chair, I expected to receive an official offer letter within a couple of weeks. However, it has now been nearly four weeks since I was notified that the dean’s approval is pending, and I am starting to feel anxious about the delay, especially since the verbally agreed start date is approaching very soon. I am contemplating whether I should reach out to the department chair or search committee for a progress update, or if I should continue to wait. Do you think it's okay for me to send a follow-up email? Also, is it normal for the dean's approval process to delay the official offer letter this long?