In a search for \(\omega\) mesic states, the production of \(\omega\)-mesons in coincidence with forward going protons has been studied in photon induced reactions on \(^{12}\)C for incident photon energies of 1250 - 3100 MeV. The \(\pi^0 \gamma\) pairs from decays of bound or quasi-free \(\omega\)-mesons have been measured with the CBELSA/TAPS detector system in coincidence with protons registered in the MiniTAPS forward array. Structures in the total energy distribution of the \(\pi^0 \gamma\) pairs, which would indicate the population and decay of bound \(\omega~^{11}\)B states, are not observed. The \(\pi^0 \gamma\) cross section of 0.3 nb/MeV/sr observed in the bound state energy regime between -100 and 0 MeV may be accounted for by yield leaking into the bound state regime because of the large in-medium width of the \(\omega\)-meson. A comparison of the measured total energy distribution with calculations suggests the real part \(V_0\) of the \(\omega~^{11}\)B potential to be small and only weakly attractive with \(V_0(\rho=\rho_0) = -15\pm\) 35(stat) \(\pm\)20(syst) MeV in contrast to some theoretical predictions of attractive potentials with a depth of 100 - 150 MeV.