We present the analytical solutions of the Klein–Gordon equation with the modified Rosen–Morse potential within the framework of an approximation to the centrifugal potential for any l state. The explicit expressions of bound state energy spectra and eigenfunctions are obtained. We also numerically solved the Klein–Gordon equation without any approximation to the centrifugal term for the same potential and compared the numerical energy levels with those of the approximate analytical results. It is found that the approximate analytical results of energy eigenvalues are in good agreement with those numerically obtained.