Many investigations have been made about of Non-Newtonian calculus and superposition operators until today. Non-Newtonian superposition operator was defined by Sağır and Erdoğan in [9]. In this study, we have defined *- boundedness and *-locally boundedness of operator. We have proved that the non-Newtonian superposition operator $_{N}P_{f}:c_{_{0,\alpha }}\rightarrow \ell _{1,\beta }$ is *-locally bounded if and only if f satisfies the condition (NA₂′). Then we have shown that the necessary and sufficient conditions for the *-boundedness of $% _{N}P_{f}:c_{_{0,\alpha }}\rightarrow \ell _{1,\beta }$ . Finally, the similar results have been also obtained for $_{N}P_{f}:c_{\alpha }\rightarrow \ell _{1,\beta }$ .