With the aid of the decomposition method and the Lyapunov direct method, stability of linear gyroscopic systems with switching and a constant delay in positional forces is investigated. The cases of synchronous and asynchronous switching are studied. The efficiency of the application of the Razumikhin approach and Lyapunov—Krasovskii functionals for the stability analysis of such systems is compared. The results of a numerical simulation are presented to illustrate the obtained theoretical conclusions.