Although it may seem part of their nature, you can train a dog not to chase cats. This is a guide about training a dog not to chase cats.
How do I stop my dog from chasing cats?
By Bonnie J from Prineville, OR
Here is a nice article about introducing Dogs into homes with cats and chasing issues.
Is the cat your own or a stray that comes into the yard? If it's a stray, the dog may just be trying to either protect its territory, or maybe (like mine) they simply want to play with the cat. Altho the cat for sure doesn't understand this!
But if it's your own cat that's a whole different ball game. Here's how I have always introduced a new pet (cat or dog, kitten or puppy) to my 2 rambunctious dogs.
I have a fairly large size wire pet cage. I set it up in the middle of the living room with the "newbie" inside. If it's a cat or kitten I set up a small litter box, and for either cats or dogs, I set up food and water in there. Of course with a new dog or puppy I put my 2 in the bedroom while I take the new one outside to do his business.
I keep the newbie in the cage tho till there is no longer any sign of distress on its part, and till my 2 have accepted it and are no longer constantly wanting to be poking their noses into the cage. This can be anywhere from a couple of hours to 2-3 weeks. Just depends on the animals. But I've never had this system fail. It forces them to be together without the worry of fights and anybody getting hurt. Good luck with yours!
Our dogs have responded fairly well to the "Leave It" command when they begin to bother the cats. That only works when we are present to give the command. After a while they might cease in their attempts to bother cats. Our cats mainly live in the large finished basement w/ their own private picture window, so don't feel sorry for them, they are quite content. Sometimes the dogs are just curious & sniff the cats or want to play w/ them. It might help to give the dog plenty of attention in the presence of the cat, or pay attention to them simultaneously if you can. There is some valuable info on the internet related to this topic.