Before reporting offically that you have a problem with your manager, and also it depends how is your relationship with your manager, you could simply ask for advice from him. And i'm talking about an honest advice, not a snarky way to involve your manager. So if you want to give it a go, make it clear to your manager that you are only solliciting for advice at this point.
Depending of your manager the answer might be something like :
- "Just shrug if off"
- "Continue like this, keep me in check if it become worst.
- "Don't let yourself get trampled when he tries to"
This will also give you an idea on how your manager see this kind of issue and how he will supports you (or not) in case you get really fed up with that and want to escalate it.