I've been in this dilemma, and required permission from my employer to apply for a job with a vendor. It was in the contract they had with us, and was bi-lateral. I understood that, and the reasoning. It was a big-league once every 10 year opportunity. I asked for permission to apply, and it was granted with zero hostility. In fact it was "Hope you get it man, it's a life changer." Anyone in this building would have jumped at the chance to be a candidate.
I actually ended up withdrawing my application as I was promoted within, and took the guaranteed promotion. But the end of that story was, the company valued me enough to advance me within once they found out I wanted to spread my wings. I have seen this happen a lot. So it may work out for you.
However, if there is no working relationship, or contract between the two companies, it is asinine for a company to request you potentially risk losing your job, to simply have the opportunity to apply. That is a very poorly managed company, and says something about them right out of the gate.
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