A data set corresponding to an integrated luminosity of $9 \text{fb}^{-1}$ of proton-proton collisions collected by the LHCb experiment has been analysed to search for $B_c^+ \to D^{(*)+}_{(s)} \overset{\scriptstyle (-)}{D}{}^{(*)0}$ decays. The decays are fully or partially reconstructed, where one or two missing neutral pions or photons from the decay of an excited charm meson are allowed. Upper limits for the branching fractions, normalised to $B^+$ decays to final states with similar topologies, are obtained for fourteen $B_c^+$ decay modes. For the decay $B_c^+ \to D_s^+ {\overline{D}}^0$, an excess with a significance of 3.4 standard deviations is found.