The bound state energy eigenvalues and the corresponding wave functions of the “deformed” Rosen-Morse and modified Rosen-Morse potentials have been obtained by the Nikiforov-Uvarov method [Büyükkılıç et al. Theor. Chim Acta. 98, 192 (1997). The “deformed hyperbolic functions” that were introduced for the first time by Arai [J. Math. Anal. Appl. 158, 63 (1991)] have been used. The energy eigenvalues of these “deformed hyperbolic molecular potentials” bring up the criteria for the shape invariance of the potentials according to the deformation parameter q, as well as bringing a bound for the value of the parameter.