Employing Coulomb and proximity potentials, cluster-decay half-lives (Tc,th1/2(s)) of even–even nuclei had been calculated and, the obtained results are compared with the experimental ones. To investigate cluster radioactivity in superheavy nuclei (SHN), we determine the required disintegration energies (Qc-values) utilizing periodic-orbit theory within microscopic-macroscopic formalism and hence, predict the logarithmic values of Tc,th1/2(s) for even–even SHN with Z=122, 124 and 126. Among various isotopes of daughter nuclei, products produced with minimum value of Gibbs free energy are chosen for our investigation. Moreover, study of the branching ratios would make us to find the dominant mode of decay among α-decay, spontaneous fission (SF) and cluster-decay in these SHN. Mostly, the emissions of Kr-, Sr-, Zr- and Mo-clusters from these superheavy nuclei are found to be the dominant modes of decay which can be attributed to strong shell effects in the neighborhood of Z=82 region. Our results are in agreement with those obtained using UDL method except for few cases. We believe that this work can play a significant role in the study of decay properties of superheavy nuclei.