When they're allowed to interact, does the dog want to sniff? chase? lie down and ignore the cat? Does the cat bristle? attack? rub or lick the dog? Does the cat stalk with tail twitching angrily or playfully?
It's so hard to tell what's going on without seeing it. The cat might just be initiating play.
We took in a stray and he (JD) and the dog (Grace) tried so hard to play. She'd play bow and feint to the side and JD would look at her like she was nuts. He'd run up next to her and try to wrestle or nibble at her elbows, and Grace would back away like he was possessed of some demon! So many signals, so little communicatin!
But they were both just trying to initiate play.
So, if you can determine if the cat is bored and wants to play, or whether she's decided the dog is a danger, that would go a far way toward figuring out what steps you need to take to keep everyone safe.