Last year, I reached out to several potential PhD advisers as I prepare for my PhD applications. One of the potential supervisors (let's call him supervisor A) with ties to the industry responded and offered me a short-term research position at their company (which I accepted) while I prepare for my PhD application at university A.
However, I was recently contacted by another professor (supervisor B) at university B. I was able to interview with his team and successfully received a PhD offer. I am inclined to study at university B, but am hoping for a scenario where I can still work with supervisor A as an industry partner or co-supervisor. What is the proper etiquette in informing supervisor A about my other PhD offer at university B?
Note that at this point, I have not formally committed to a PhD program yet despite having accepted the short-term research position with supervisor A. But there is the expectation that I would apply at university A and pursue my PhD there (to be fully funded by supervisor A).
Will I be burning bridges if I decide to pursue my doctorate at university B? Considering that PhD is a major commitment, I would hate to turn down the opportunity out of decorum/politeness. What is the best way to bring this up with supervisor A? Would it make sense for me to invite him to be my co-supervisor at university B?
Edit: I'm particularly curious about the ethics of switching gears when I've already given the impression to one supervisor that I'll be pursuing a particular PhD program. What's the best way to bring this up (email, direct 1/1 conversation) and how should I word it so as not to burn bridges?