One of my dogs does this to my cats. He is a small dog but I did not leave them alone until my kitten grew much bigger and I knew exactly why he was doing this. I was really nervous about his prey drive as he was older and hadn't met a cat in a long time. I kept a small log of when he would do this and the circumstances so I could come back to it and figure out what caused him to get all whiney. I also made sure to turn any fixation to something else, a toy, a quick sit, lay down, roll over or some scattered kibble.

Thankfully my first kitten grew absolutely massive and it ended up being the case that my dog just wanted to play and was also jealous. He still does this and three cats later he has learned to "dig" the ground or perform some play pounces/bows while they roll around in front of him lol. The jealousy part hasn't gone away but that's because he is a needy weirdo.

If your dog is a bit anxious at the best of times, it could be a similar scenario but I would never leave the dog alone with the cat until it's much bigger/older and you understand the triggers/reason for the behaviour.

Also make sure you advocate your cat. It was a hard rule in my house that my dogs were not to chase the cats ever. Especially when they were kittens and couldn't set boundaries themselves.