My favorite moment between Bruce and Gordon comes from the No Man's Land story from the '90s, where it comes to the point that Bruce feels that he and Jim are close enough friends that he has to tell him who he is. While talking to Jim, he starts to remove his cowl, and after he does so, he looks up to see Jim facing away from him. He didn't want to know, and even though it would be so easy to turn around and see who Batman really is, he refused. Batman put his cowl back on, and Jim knew that Batman really was his most trusted ally, and even friend, if he felt strongly enough to do something like this
Depends on the iteration in most iterations he doesn't know and doesn't suspect he is Bruce but in others he either eventually finds out or suspects he is Bruce
