In this study, constructal theory is used to analyse the radial and branching configurations of highly conductive incomplete inserts embedded in a disc for cooling purposes. The thermal conductivities are considered temperature-dependent which are varied linearly and ascending. Applying the Kirchhoff transformation, the resulting nonlinear partial equations are transformed to linear ones which are more suitable to solve, analytically. Furthermore, the effect of temperature-dependency of the thermal conductivities is examined and compared with the case with constant thermal conductivities under different conditions where a decrease in heat resistance is observed. However, the effectiveness of this factor in decreasing the thermal resistance is dependent upon other parameters. The effect of considering variable cross-section for the highly conductive material is combined with the temperature-dependency of the conductivities where it is concluded that under certain circumstances where the effect of temperature-dependency of the conductivities is weak, the variable cross-section can be used to reduce the thermal resistance, efficiently. Finally, since the proposed analytical solution is accompanied with some approximations, the problem is also solved numerically to verify the solution where an acceptable agreement is observed.