As far as I could tell, the President could decide to do whatever he wants with the "football" (in terms of where it is, anyway), but it would be useless if not near him and so would defeat the entire purpose. The football by itself is just a communications device and the relevant documentation for emergency options (nuclear and otherwise). It's less talked about, but the Vice President also has his own copy.1 The actual authentication codes (not launch codes!) to verify the President's identity are carried on a separate index card which the President carries with them. The Vice President also has codes, for the same purpose - they authenticate the VP to whoever's at the other end of the football's communication link, thereby allowing him to order a strike if he has the authority to do so (i.e. is Acting President).
The briefcase has been with Presidents in both Soviet Russia and the Vatican.2,3 It's also been left behind or misplaced on occasion, although never for long. The aide doesn't necessarily have to stand next to him at all times, but just be close by for ease-of-access. Presumably, when he's actually asleep on a trip, the aide carrying it is in the next room with the rest of his bodyguard detail. So it would be unheard of for Obama to order it out of his immediate vicinity, although he could likely instruct the aide to hang back somewhat to avoid being in photographs.