In recently proposed framework of non-holomorphic modular symmetry introduces the concept of negative and zero modular weight of Yukawa couplings. These Yukawa couplings are function of complex modulus τ, which is responsible for the CP asymmetry produced during leptogenesis. In this work, we restrict the τ on the fixed points of modular symmetry rather than its fundamental domain in such manner Yukawa couplings are also get fixed. We have adopt this framework and propose a type III seesaw mechanism. The model is tested against neutrino oscillation data through a χ2 analysis using NuFIT 6.1. To test the stability of these predictions, we also analyze regions near each fixed point by introducing a deviation τ → τfixed(1+ϵeiϕ) with ϵ ∈ (0, 0.1) and ϕ ∈ (−π, π). Our results show that certain fixed points, along with their nearby regions, are capable of producing viable neutrino phenomenology while also generating the observed baryon asymmetry of the Universe.