Here's the entire quote:
Fox: What can I do for you, Mr. Reese?
Reese: You asked me to do the diligence on the LSI Holdings deal again. I found irregularities.
Fox: Their CEO is in police custody.
Reese: Not with their numbers. With yours. Applied Sciences, a whole division of Wayne Industries disappeared, overnight. So I went down to the archives and started pulling old files.
He pulls out a folded blueprint. Slides it across the desk.
Fox picks up the piece of paper. Unfolds it. It's an old BLUEPRINT. The image is unmistakeable: THE TUMBLER.
Reese (cont'd): Don't tell me you didn't recognise your baby pancaking cop cars on the evening news. But now you've got the entire R and D department burning cash, claiming it's related to cell phones for the army. What are you building him now? A rocket ship?
Reese (cont'd): I want ten million dollars a year. For the rest of my life.
Fox looks at him. Even. Folds up the blueprint.
Fox: Let me get this straight. You think that your client, one of the wealthiest and most powerful men in the world, is secretly a vigilante who spends his nights beating criminals to a pulp with his bare hands... (deadpan) And your plan is to blackmail this person?
Reese stares at Fox. Who smiles. And slides the blueprint across the desk.
Fox: Good luck.
Reese looks at it. Then at Fox. Swallows. Slides it back.
I had to type this out, copying from the screenplay, so there may be small errors.
No, he doesn't show that he has anything directly identifying Wayne as Batman. But he does at the very least have evidence that Wayne and Lucius were supporting him. Given Batman's lone-wolf nature, it's not a stretch to think Wayne IS Batman. And he may have had other evidence we don't see.
Lucius is a smart guy. He knows that the best way to deal with it is to actually encourage the belief that Wayne is Batman. Because if he is, he's trying to blackmail Batman and that's a colossally bad idea. But he doesn't say anything to admit it- on the contrary, you could take it to be a healthy scepticism.
Reese's evidence could have been a hell of a threat. Lucius took the only potentially clean way of dealing with the problem, by making the guy too afraid to do anything about what he'd found. It wasn't a perfect solution by any means, because if he'd been less easily rattled he would have stood his ground and Lucius would have had no choice but to accede to his demands.
In the end, Reese must have realised that if he went public, he could at least sell the information for a fair bit and the publicity would mean Wayne couldn't touch him, because he did attempt to go public, causing The Joker to put a hit on him.
I suspect after that he would have kept it quiet, because a threat from someone like The Joker carries much more weight because he has no foolproof defence against his thugs, even if the man himself is in jail.
EDIT: After comments showing that it plays out slightly differently in the film itself, I've updated it while listening to the dialogue in the movie.