The convection heat transfer inside a tube filled with a porous material under the constant heat flux thermal boundary condition which is widely used in practical applications is studied in the present article. The Darcy–Brinkman–Forchheimer model is used to cover a wide range of working mediums from clear fluid flow to slug flow (Darcy flow). The case of temperature-dependent thermal conductivity is considered in the present study and the corresponding Nusselt number is analytically obtained using perturbation techniques for the first time. The change in thermal conductivity with respect to temperature occurs at high-temperature applications wherein high-temperature variations exist as well as the radiation heat transfer. A linear model for the thermal conductivity variation with temperature is considered in the present study. The obtained profile for the Nusselt number can be used for quick calculations as well as validation of numerical and experimental studies, especially at high temperatures wherein the experimental studies are accompanied by higher uncertainties. The results show that the Nusselt number increases linearly with the linear increase in the thermal conductivity and as well the heat transfer rate. Furthermore, results show that the Nusselt number (and the heat transfer rate as well) shows more augmentation to the thermal conductivity enhancement due to the temperature-dependent nature of thermal conductivity (especially arising from the radiation heat transfer) in comparison with the clear fluid flow case.