In this article, we examine the prospect of a first-order phase transition with a Y=0 real SU(2) triplet extension of the Standard Model, which remains odd under Z2, considering the observed Higgs boson mass, perturbative unitarity, dark matter constraints, etc. Especially, we investigate the role of Higgs-triplet quartic coupling considering one- and two-loop beta functions and compare the results with the complex singlet extension case. It is observed that, at one-loop level, no solution can be found for both, demanding Planck-scale perturbativity. However, for a much lower scale of 104 GeV, the singlet case predicts a first-order phase transition consistent with the observed Higgs boson mass. On the contrary, for the two-loop beta functions with one-loop potential, both the scenarios foresee a strongly first-order phase transition consistent with the observed Higgs mass with upper bounds of 310 and 909 GeV on the triplet and singlet masses, respectively. This mass bound shifts to 259 GeV in the case of a triplet with the inclusion of two-loop contributions to the effective potential and the thermal masses with two-loop beta functions, consistent with Planck-scale perturbativity and the observed Higgs boson mass value. This puts the triplet in apparent contradiction with the observed dark matter relic bound and, thus, requires an additional field for that. The preferred regions of the parameter space in both cases are identified by benchmark points that predict gravitational waves with detectable frequencies in present and future experiments. © 2023 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the "https://creativecommons.org/licenses/by/4.0/"Creative Commons Attribution 4.0 International license.