In this manuscript,the traversable wormhole solutions have been explored in modified f(R,G)gravity by taking into consideration the model f(R,G)=R+γG2,where R is the Ricci curvature scalar and G is the Gauss-Bonnet term.An acceptable form of the redshift function has been incorporated which is a non-constant in nature along with the implementation of those already defined in literature,the two shape functions namely b(r)=r/exp(r-r0)and b(r)=r0log(r+1)/log(r0+1).It is shown by studying the energy condition and through the graphical analysis that the null energy bounds for the effective energy-momentum tensor are generally violated for the presence of the ordinary matter in modified f(R,G)gravity.Energy conditions associated with the matter content threading the wormhole geometries are evaluated and in general are found to favor the null energy conditions in the vicinity of the wormhole throat,the existence of the non-exotic wormhole geometries threaded by the matter has been confirmed under this gravity.