Delayed start date due to internship
I am thinking about whether or not to ask a (potential) future employer about delaying a start date by 3 to 4 months, in order to complete a short-term internship. The internship would be a valuable opportunity to round out my skills before starting a more permanent position in a slightly different area; the two employers do not compete with each other. How open do you think future employers would be to this type of situation?
First and importantly if I was the hiring manager I wouldnt have a problem with you asking the question.
How open I would be to the idea depends on the specific situation and if the skills that you will learn also will have a value to us as a company. Generally 3-4 months is probably longer than I would like to wait but again depending on the specific situation it might be possible to find a compromise ie let you start for example 2 months later.
In my experience in industry (especially biotech), a job is open because they want someone to start literally yesterday and hit the ground running. I'd be hesitant to mention this during the interview/post-interview process, but if you've already been offered the position it can't hurt to ask the question.
You also have some negotiation room here as PG stated. If the internship will give you skills/techniques that are relevant to the new job, then it strengthens your case. In general the job wants to know "what can you do for us" and isn't looking to help you out (at least until you get there and they've made an investment in you!). If possible, you might be able to arrange a part-time situation with both positions with 2 days a week at the internship and 3 days at the job, especially if you're willing to put in >8 hours/day.
An employer's willingness to honor this type of request is probably also going to depend on what level you're being hired for. If you're being hired for a position that requires some experience and for which there is a specific need, you're more likely to receive some push-back. If you're being hired for an entry-level position that's part of a standard hiring program, they'll likely be more amenable to the idea. Just be aware that if you're receiving the offer before you've received your PhD, they're not going to be too happy to wait 2-4 months for you to defend, then another 3-4 months for you to finish an internship (unless is really brings some value to the company).