We investigate a problem of convection in local thermal non-equilibrium (LINE) porous media. This is where the solid skeleton and the fluid may have different temperatures. However, due to increasing interest in thermal wave motion, especially at micro-nano-scales and particularly in solids, we analyse a model for thermal convection in a fluid saturated Darcy porous medium allowing the solid and fluid parts to be at different temperatures. This theory employs thermodynamics for the fluid based on Fourier's law of heat conduction, whereas for the solid skeleton heat is transferred by means of the Cattaneo heat flux theory. Under appropriate conditions oscillatory convection is found whereas for the standard LINE Darcy model this does not exist. In this article we concentrate on a transition which is found when the heat transfer interaction coefficient reaches a critical level and this leads to an exchange of stability for a certain porosity level. (C) 2015 Elsevier Ltd. All rights reserved.