In order to explore the spectrum of hidden-charm scalar and tensor resonances, we study meson-meson scattering with $JPC=0++,2++$ in the charmonium energy region using lattice QCD. Employing a light-quark mass corresponding to $m\pi \approx 391$ MeV, we determine coupled-channel scattering amplitudes up to around 4100 MeV considering all kinematically relevant channels consisting of a pair of open-charm mesons or a charmonium meson with a light meson. A single isolated scalar resonance near 4000 MeV is found with large couplings to $DD¯$, $DsD¯s$ and the kinematically closed $D\ast D¯\ast$ channel. A single tensor resonance at a similar mass couples strongly to $DD¯$, $DD¯\ast$ and $D\ast D¯\ast$. We compare the extracted resonances to contemporary experimental candidate states, previous lattice results and theoretical modeling. In contrast to several other studies, we do not find any significant feature in the scalar amplitudes between the ground state $\chi c0(1P)$ and the resonance found around 4000 MeV.