Abstract: In this paper, we consider the possibility of generating the observed baryon asymmetry of the Universe via leptogenesis in the context of a triplet fermion-mediated type-III seesaw model of neutrino mass. With a hierarchical spectrum of the additional fermions, the lower bound on the lightest triplet mass is similar to 1010 GeV for successful leptogenesis, a couple of orders higher than that of the type-I case. We investigate the possibility of lowering this bound in the framework of two-Higgs-doublet models. We find that the bounds can be lowered down to 107 GeV for a hierarchical spectrum. If we include the flavor effects, then a further lowering by one order of magnitude is possible. We also discuss if such lowering can be compatible with the naturalness bounds on the triplet mass.