A recently proposed scenario for baryogenesis, called post--sphaleron baryogenesis (PSB) is discussed within a class of quark--lepton unified framework based on the gauge symmetry SU(2)_L x SU(2)_R x SU(4)_c realized in the multi--TeV scale. The baryon asymmetry of the universe in this model is produced below the electroweak phase transition temperature after the sphalerons have decoupled from the Hubble expansion. These models embed naturally the seesaw mechanism for neutrino masses, and predict color-sextet scalar particles in the TeV range which may be accessible to the LHC experiments. A necessary consequence of this scenario is the baryon number violating Δ B=2 process of neutron--antineutron (n-\bar{n}) oscillations. In this paper we show that the constraints of PSB, when combined with the neutrino oscillation data and restrictions from flavor changing neutral currents mediated by the colored scalars imply an upper limit on the n-\bar{n} oscillation time of 5 x 10^{10} sec. regardless of the quark--lepton unification scale. If this scale is relatively low, in the (200-250) TeV range, τ_{n-\bar{n}} is predicted to be less than 10^{10} sec., which is accessible to the next generation of proposed experiments. ; Comment: 22 pages, 3 tables, 10 figures; clarification added on the baryogenesis calculation; version accepted for publication in Phys. Rev. D