Could the pope adopt a child?

In conversations with my son (topic irrelevant), he mentioned the idea of being the son of the Pope. I told him that would not be possible (celibacy vow), but he immediately countered: "What if he adopted?". I was not able to provide a satisfactory answer. I've seen the same question about priests (needing permission from the Church) or having a child before being ordained (is OK). But not about the current pope (or a future one). Would there be any obstacle to this scenario?