This can be a consequence of the anatomy of your dog. Some Chihuahuas have eyes that slightly bulge which leads to tears spilling over the lower lid, rather than draining through the nasolacrimal duct into the nose. However, there may also be an obstruction of the nasolacrimal duct, especially if this is a new development. Have your veterinarian take a look at your dog’s eyes to determine if treatment is required.