The density response of supercooled glycerol to an impulsive stimulated thermal grating (q=0.63 microm(-1)) has been studied in the temperature range (T=200-340 K) where the structure rearrangement (alpha relaxation) and the thermal diffusion occur on the same time scale. A strong interaction between the two modes occurs giving rise to a dip in the T dependence of the apparent thermal conductivity and a flattening of the apparent alpha-relaxation time upon cooling. A nonequilibrium thermodynamic model for the long time response has been developed. The model is capable to reproduce the experimental data and to explain the observed phenomenology.