Ask yourself: "Do I deserve a promotion?"
I've applied to jobs internally before. I doubt your manager will let you go but truthfully, it's always best to be transparent with your manager. If you're not satisfied with your roles and responsibilities, your salary, your productivity, or processes within your team, you should voice your concern rather than not saying anything and jumping ship.
By the way, your manager saying there is no promotion opportunities is kind of misleading because it makes it sound like there is nothing he can do, which is false. But it can be true if HR refuses to provide that opportunity. If you manager truly saw a reason to be able to promote you, he would go to HR and practically demand it.