In this study, a Lattice Boltzmann Method (LBM) has been developed to calculate the distribution of a scalar quantity, like temperature, in a natural convection flow field under the condition of varying fluid thermal conductivity. The standard form of an LBM usually considers the fluid properties to be constant without any source term in conservation equations. The model developed is to account for variation of thermal conductivity with temperature in the presence of an external heat source. The proposed model has been examined against various case studies. It is shown that it is capable of modeling the extremely nonlinear problems. To magnify the nonlinear term in the natural convection case of under study, the radiation and other thermal sources have been used. The multiple relaxation time scheme has been applied to assure the solution stability. Using Chapman-Enskog analysis, the error associated with the proposed model has been estimated. The part of error which was not due to variations in the fluid properties, may be eliminated by introducing a correction term in higher order terms in Chapman-Enskog analysis. In addition, it has been shown that the correction term associated with the fluid conductivity variations, create an error of second order in terms of Knudsen number and is negligible. The present LBM model has an error of the second order of magnitude with respect to time.