Motivated by current experiments on searching for neutrinoless double beta decays, we study the phenomenological consequences and testability of the constraint relation “Mii=Mjj+Meμ+Meτ”, for i,j=e,μ,τ, on the magic neutrino mass matrix elements. This hypothesis is inspired by the μ−τ symmetry and gives rise to six different Majorana neutrino mass textures, including two textures one-zero. These obtained textures are diagonalized by the well-known trimaximal mixing which provides nonzero and nonmaximal reactor and atmospheric angles, respectively. Depending on the predominance of the e-row or μ-τ block, the preferred neutrino mass hierarchy and the lower limits on the |Mii| effective Majorana masses for each magic matrix are determined. Furthermore, in light of the global-fit results of neutrino oscillation parameters as reported by NuFIT 5.1, we also study the testability of these textures at the proposed neutrinoless double beta decay experiments such as KamLAND-Zen, GERDA Phase-II and nEXO, by incorporating the effective Majorana neutrino masses |Mij| that are generated through these neutrino mass matrices.