Both 13C 16O and 22Ne 25Mg reactions perform a cosmic role in the production of neutrons in AGB stars, which significantly contributes to the nucleosynthesis via the s-process. The astrophysical S-factor for both reactions has been calculated in this research, utilizing EMPIRE code and depending on two parameter sets for the optical potential: McFadden and Satchler (MFS), and Avrigeanu and Hodgson (AH) for the non-resonant region of the spectrum, and over a temperature range . The extrapolated S-factor at zero energy is derived to be  and  for 13C 16O,  and  for 22Ne 25Mg from using MFS and AH parameter sets respectively, and they show a reasonable agreement with the most recommended value. The differences in the S-factor, S(E), values obtained from these two adopted parameters set, were attributed to the variations of the real potential term's diffuseness parameter that affecting the reaction cross section, hence S-factor, and specifically at low energy region. Moreover, the present results imply an influential enhancement of the rates by the electron shielding effect at the low-temperature region  in which 13C 16O reaction activated, and especially on 22Ne 25Mg reaction. In addition, for both adopted reactions and overall selected temperature range, the reaction rates using  values based on MFS showed acceptable results with other previous compilations and reference libraries. While that obtained from AH exceed all other compilations even though the resonance contributions are currently unconsidered.