[en] A study is presented of the reactions K⁺ p → (K⁺ω)p at 8.25 and 16 GeV/c and K^- p → (K^- ω)p at 10 and 16 GeV/c and comparison is made with K⁺ results at 10 GeV/c and K^- at 7.3 GeV/c. The (K⁺ω) and (K^-ω) mass spectra both present a strong enhancement very near threshold, while a second peak at approx. 1.7 GeV is evident only with incident K^- at the lower energies. The threshold peak has very weak energy dependence and is mostly due to the 1⁺ S state which is produced conserving s-channel helicity. It is suggested that this is another decay mode of the resonance Q₁(1290) known to decay mainly into Krho. The ratio of the Q₁ coupling constants to the Kω and Krho decay channels, Rω = g²sub(K)ω/g²sub(K)sub(rho) is determined to be 0.21 +- 0.04. The enhancement at 1.7 GeV is predominantly, but not exclusively, due to the 2^- state. While the K⁺ and K^- induced reactions give basically similar results, small differences are observed that can be qualitatively explained in the framework of the Deck model. (orig.)