Probably something in the contract, but I'd say the chances of him leaving New Orleans are almost nil. They're kind of an anomaly. A team on the rise with an aging QB. If they had drafted Brees' replacement recently, you could see it happening. Or if the Saints were a team on the decline. Brees is the key to them remaining in the hunt and there's no way they don't re-sign him. Brees will probably sign a very team friendly deal.
See, these are the kind of QBs we need. Brady, Brees. Guys who've been around so long, offensive coordinators adjust to THEM, not vice versa. Guys who are such a big part of the community that they wouldn't even conceive of leaving. And will take less money to help the team remain competitive, because they're smart enough to know its in their best interest. Until that dark day in August of '16, I believed Teddy would eventually be that kind of QB.