I've seen a source say his father was Bheemana Bhatta (భీమన భట్ట).
https://istampgallery.com/adikavi-nannaya/
Also, according to Wikipedia he was born in Tanuku, West Godavari. He died in Rajahmundry. Maybe you could his birth date as c. 1000?
https://en.wikipedia.org/wiki/Nannayya / https://te.wikipedia.org/wiki/%E0%B0%A8%E0%B0%A8%E0%B1%8D%E0%B0%A8%...
Thank you for this