We investigate thermal convection in two horizontal layers of miscible fluids with an initial state in which the fluids are at rest and the temperature has a linear continuous vertical distribution, but the concentration takes different constant values in each layer. Because the initial layered state is not a stationary state, most theoretical methods available for immiscible fluids are not applicable to the miscible case. Here we propose a new model of the thermal convection of miscible fluids, which is devised in such a way that the horizontal average of the width of the transitional layer is kept constant. Both the stability of the stationary state in the linear stage and the asymptotic attractors in the nonlinear stage of the model closely describe the transition of the convection pattern in the original system, which is associated with the change of the width of the transitional layer.