Tricky question. There have been many pieces of research written by vets and cat experts and there hasn’t been a joint conclusion. If you want to be totally safe, consider buying a bib for your cat. The bib is usually a cover for your standard cat collar.
Cat bibs generally come in bright colors as a visual warning for birds, hang loosely from the cat chest and work by gently interfering with the precise timing and coordination a cat needs for successful bird catching.