We study the in-medium masses of the charmonium states and in the nuclear medium using the QCD sum rule approach. These mass modifications arise owing to modifications of the scalar and the twist-2 gluon condensates in the hot hadronic matter. The scalar gluon condensate and the twist-2 tensorial gluon operator in the nuclear medium are calculated from the medium modification of a scalar dilaton field introduced to incorporate trace anomalies of QCD within the chiral SU(3) model used in the present investigation. The effects of isospin asymmetry, density, and temperature of the nuclear medium on the in-medium masses of the lowest charmonium states and mesons are investigated in the present work. The results of the present investigation are compared with the existing results on the masses of these states. The medium modifications of the masses of these charmonium states ( and ) seem to be appreciable at high densities and should modify the experimental observables arising from the compressed baryonic matter produced in asymmetric heavy-ion collision experiments at the future facility of Facility for Antiproton and Ion Research, GSI.