In the season 4 of Bleach anime, one of the main characters state that the bounts are born or created in the soul society.
Are souls born in the soul society as soul reapers? Or if everyone was once a human and died and came to the soul society? Or if everyone was somehow created like the bounts?
Bounts are non-canonical entities.
We should begin with this information because it means your question isn't actually relevant to Bleach's canon (although you didn't know that, obviously, and it isn't your fault if you're an anime-only watcher).
Bleach's lore has only gotten more developed and complicated during the last years, especially thanks to the Can't Fear Your Own World light novel (which is generally accepted as --somewhat-- canonical since written with Tite Kubo's, Bleach's creator, complete approval).
In short, before the three different dimensions were split apart, there only existed "souls" -- shinigami-like souls made out of spirit particles. When the worlds were split, the reincarnation cycle was created -- with a balanced amount of souls between the three dimensions to maintain balance.
So, to answer your question, a third of all souls was transformed into Hollows, a third was sent to the Spirit Realm and a last third became humanity. But each and every soul stems from the "proto-souls" entities from before.
