So I am currently employed at an Information Technology/ Help Desk internship through my school's work study program, and in order to make my transition out as smooth as possible, my boss has asked me to train my replacement for (ideally) a month. I have no real problem doing this, as I am not located far from either work or school and the person I had in mind to replace me is a friend of mine.
However, the process of actually transitioning out is a little more complicated. I do not have another job secured per se, but I can continue to tutor at the school at the same hours and pay until I graduate. Since I'd rather not be without the money, I'm thinking the best way to do this is leave the internship early, help train the replacement, and tutor until I can find a job.
This seems to be the safest back-up option to me, but I would like to start interviewing ASAP. So my question then is this: Suppose I interview well and manage to get an offer in the same week as I begin training my replacement. Would I be unreasonable in proposing a start date a month out?