<p>This paper introduces a novel model of a spherical solid thermoelastic material with variable thermal conductivity, constructed under the More-Gibson-Thompson theorem of thermoelasticity. A dimensionless system of governing equations based on the Laplace transform method has been applied to a thermoelastic, isotropic, homogeneous solid sphere thermally loaded by a ramp-type heat flux on its surface, assuming that the surface has no volumetric strain. The Laplace transform inversions have been computed numerically using the well-known Tzou’s iteration method. The singularity at the centric point inside the sphere has been reduced by implementing the L’Hopital rule. The numerical results for the temperature increment, volumetric strain, and invariant-average stress distributions are shown in the figures and discussed. The assumption of thermal conductivity variability significantly influences all understudied functions and the behaviours of the thermomechanical solid sphere. The ramp-time heating parameter has a major impact on the distributions of temperature increment, volumetric strain, and invariant average stress. So, the ramp-time heat parameter tunes the thermomechanical waves through the thermoelastic material.</p>