@curtis697 You’ve gotten some good advice and stories on how to hopefully fix this issue, so I’m offering a perspective from someone who’s lived with the chasing behavior being unresolvable:
I’ve got a sighthound and a cat. Some sighthounds do great with cats indoors, especially if they’ve been raised with one. Unfortunately ours isn’t one of those sighthounds. We have never been able to train away his behavior of trying to chase the cat, his prey drive is just way too strong. The cat is also small and has always been naturally timid, so she doesn’t swipe at him she just runs which makes him want to chase even more. No matter how hard we trained he just won’t stop chasing, he does this with any small furry animal he sees inside or outside.
So the permanent solution we settled on that keeps everyone safe and happy is to keep the two completely separate. The dog gets the downstairs, and basement and is kept from going upstairs by a tall baby gate at the base of the stairs. The cat gets the upstairs of the house to herself and has some cat shelves in the living room that are accessible from the stairs so she can safely hang with us downstairs. The dog doesn’t try to get upstairs and the cat doesn’t try to venture downstairs, they’ve both grown accustomed to what is their “territory” in the house.
When we are not at home we keep the dog in his own room downstairs with the door closed as a precaution. The room has a comfy couch for naps, a window for watching the backyard, and he’ll get a chewy or a puzzle feeder to keep him occupied until we get back.
Is it ideal? No, but it works and keeps them both safe. All of us have been living stress free for years with this setup.
Hopefully you can address this behavior through training, but if not keeping them separate is a doable solution too.