A magnon-phonon interaction model is set up in a two-dimensional ferromagnetic double square-lattice system. By using the Matsubara Green function theory we have studied the magnon life-time of the system, calculated the magnon damping curves on the main symmetric line in the Brillouin zone, compared the influences of magnetic ion optical-mode phonon and non-magnetic ion optical-mode phonon on the magnon damping of the system, and discussed the influences of the parameter variations and temperature on the magnon damping. The results show that the coupling of the optical-mode phonon and magnon plays the main role in the magnon damping, especially, the longitudinal optical-mode phonon contributes the most for the magnon damping. It is also found that the effect of the nonmagnetic ion on magnon damping is more significant than that of the magnetic ion. According to the expression -ImΣ*(1)(k)=/(2τ), the magnon life-time is discussed.