In this paper, two heat conductor models with fading memory are obtained by different approximations of Tzou's dual-phase-lag theory. For the first model, we obtain a close coincidence between the restrictions of the second law and the asymptotic stability. Otherwise, for the second model the thermodynamic restrictions appear more restrictive compared with the conditions following from the asymptotic stability. (C) 2014 Elsevier Ltd. All rights reserved.