You can apply but to accept a position will require you to have active employment authorization. This can either be done with an H-4 EAD (if eligible per USCIS rules) or you can request University to process your H-1B which is usually cap exempt. I suggest not discussing the H-4 and H-1B situation until after a job offer has been made by a Program Chairperson or Head, most Professors don't know what a specific immigration situation means and there is no point in confusing them. Remove all immigration references from your resume. If applying online answer 'yes' if asked 'are you eligible to work in the US?' and state if possible in comments 'With USCIS permission'.