Coupling between energy and momentum transport in a dilute gas subject to uniform shear flow is analyzed. Heat flux is created in the system by the action of a nonconservative external force. The results are obtained by using the Bhatnagar-Gross-Krook kinetic model. The moments of the velocity distribution function are expressed in terms of a perturbation expansion in powers of the heat field strength, the coefficients being highly nonlinear functions of the shear rate. In particular, the thermal conductivity tenser and the shear viscosity coefficient up to second-order approximation are explicitly evaluated. It is shown that the usual choice of the heat field proposed in computer simulations leads to a thermal conductivity tenser different from the one obtained in the thermal gradient problem, confirming previous results. In order to avoid this discrepancy, an alternative external force is proposed.