In this paper, we analyze the structural properties of $Z=132$ and $Z=138$ superheavy nuclei within the ambit of axially deformed relativistic mean-field framework with NL$3^{*}$ parametrization and calculate the total binding energies, radii, quadrupole deformation parameter, separation energies, density distributions. We also investigate the phenomenon of shape coexistence by performing the calculations for prolate, oblate and spherical configurations. For clear presentation of nucleon distributions, the two-dimensional contour representation of individual nucleon density and total matter density has been made. Further, a competition between possible decay modes such as $alpha$-decay, $beta$-decay and spontaneous fission of the isotopic chain of superheavy nuclei with $Z=132$ within the range 312 $le$ A $le$ 392 and 318 $le$ A $le$ 398 for $Z=138$ is systematically analyzed within self-consistent relativistic mean field model. From our analysis, we inferred that the $alpha$-decay and spontaneous fission are the principal modes of decay in majority of the isotopes of superheavy nuclei under investigation apart from $beta$ decay as dominant mode of decay in $^{318-322}138$ isotopes.
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