A continuum approach to the thermomass theory for nonlinear heat transport is developed and its compatibility with the general framework of continuum thermodynamics is investigated. The heat flux is supposed to depend on the absolute temperature together with a vectorial internal variable, which is proportional to the drift velocity of the heat carriers. A generalized heat-transport equation, which is capable to bring Fourier, Maxwell–Cattaneo–Vernotte and thermomass-theory equations as special cases is derived. Propagation of heat waves along a nonequilibrium steady state is analyzed as well.