The out-of-plane orbital magnetic moment of Co in Co/X multilayers (X=Pd, Pt, and Ni) with perpendicular magnetic anisotropy has been investigated using x-ray magnetic-circular-dichroism spectroscopy. The orbital d moment per Co atom, ${\mathit{m}}_{\mathrm{orb}}^{\mathrm{\ensuremath{\perp}}}$, is considerably enhanced by (0.15\ifmmode\pm\else\textpm\fi{}0.02)${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$ and (0.07\ifmmode\pm\else\textpm\fi{}0.02)${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$ in Co/Pd and Co/Pt multilayers, respectively, compared with a single thick Co film. No respective enhancement is found in the case of Co/Ni. This enhancement does not scale with the spin-orbit coupling parameter ${\ensuremath{\xi}}_{\mathrm{SO}}$ of Ni-3d, Pd-4d, and Pt-5d bands despite strong d-d mixing which leads to the presence of ferromagnetic moments of Pd and Pt atoms. However, a roughly linear scaling relation of the out-of-plane orbital moment ${\mathit{m}}_{\mathrm{orb}}^{\mathrm{\ensuremath{\perp}}}$ with the intrinsic uniaxial anisotropy constant ${\mathit{K}}_{\mathit{u},1}^{\mathrm{Co}}$ per Co volume is observed. This experimental finding is interpreted in conjunction with recent ab initio and tight-binding band-structure calculations as evidence for a preferential enhancement of the out-of-plane orbital moment of Co in Co/Pd and Co/Pt multilayers.