1 reason why he wouldn't want to stay in Detroit: The Tigers are not a good team
This is pretty simple. The Tigers are not a good baseball team, and it's tough to know when that will change. E-rod will be 31 next season, so he will just be entering the end of his prime. He has won a World Series in his career, but I'm sure he wouldn't mind winning another one.
The Tigers likely won't be winning a World Series any time soon. That's pretty obvious. Winning championships are why most athletes play sports, after all. If E-rod cares about winning at all, then Detroit might not be the place for him long term.
There are plenty of teams on the east coast, closer to where his family resides in Miami, that are better than the Tigers, that will be looking for starting pitchers this offseason. The Braves, Orioles, Marlins, and Rays will likely all be in on the starting pitching market this winter. All of those teams are further along than the Tigers.
It's pretty simple. If E-rod likes it here in Detroit, and winning isn't all that important to him, than he'll stay. If he wants to win another World Series sooner rather than later, he'll go elsewhere.