We investigate the in-medium masses of a D (0^–) meson and a D₀^* (0⁺) meson, and spectral functions for D and D₀^* meson channels in nuclear matter. These mesons are introduced as chiral partners in the chiral-symmetry-broken vacuum, hence they are useful to explore the partial restoration of the broken chiral symmetry in nuclear matter. We consider the linear sigma model to describe the chiral symmetry breaking and to investigate a qualitative tendency of changes of D mesons at low density. Our study shows that the loop corrections to D and D₀^* meson masses provide a smaller mass splitting at finite density than that in vacuum, a result that indicates a tendency of the restoration of the chiral symmetry. We investigate also the spectral function for D₀^* meson channel, and find three peaks. The first peak, which corresponds to the resonance of the D₀^* meson, is broadened by collisions with nucleons in medium, and the peak position shifts to lower mass due to the partial restoration of chiral symmetry as the density increases. The second peak is identified as a threshold enhancement which shows a remarkable enhancement as the density increases. The third peak is Landau damping. The obtained properties of D and D₀^* mesons in nuclear matter will provide useful information for experiments.