We present the first lattice investigation of coupled-channel D (D) over bar and D-s(D) over bar (s) scattering in the J(PC) = 0(++) and 2(++) channels. The scattering matrix for partial waves l = 0, 2 and isospin zero is determined using multiple volumes and inertial frames via Luscher's formalism. Lattice QCD ensembles from the CLS consortium with m(pi) similar or equal to 280MeV, a similar or equal to 0.09 fm and L/a = 24, 32 are utilized. The resulting scattering matrix suggests the existence of three charmonium-like states with J(PC) = 0(++) in the energy region ranging from slightly below 2m(D) up to 4.13 GeV. We find a so far unobserved D (D) over bar bound state just below threshold and a D (D) over bar resonance likely related to chi(c0)(3860), which is believed to be chi(c0)(2P). In addition, there is an indication for a narrow 0(++) resonance just below the D-s(D) over bar (s) threshold with a large coupling to D-s(D) over bar (s) and a very small coupling to D (D) over bar. This resonance is possibly related to the narrow X(3915)/chi(c0)(3930) observed in experiment also just below D-s(D) over bar (s). The partial wave l = 2 features a resonance likely related to chi(c2)(3930). We work with several assumptions, such as the omission of J/psi omega, eta(c)eta and three-particle channels. Only statistical uncertainties are quantified, while the extrapolations to the physical quark-masses and the continuum limit are challenges for the future.