It’s a near certainty there were cats nearby when Jesus was born. Where there is food for livestock there are mice and where there are mice, there are cats. Cats were already domesticated in the time of Jesus. (As much as cats can ever be domesticated, anyway.)
Cats witnessed the birth of Jesus, perhaps rubbing up against Joseph, as the baby was born.
Here are a few images I’ve collected to give us an idea of what the birth of the Christ child must have looked like, with cats nearby.
(I’ve done my best to credit the original photographers. Many photo credits are difficult to track down. Let me know if I need to make a change to a photo credit.)