Stability plays a significant role in observing Fermi-Pasta-Ulam-Tsingou(FPUT) phenomenon in experiment successfully. However, few relevant studies have been documented in the literature. The primary object of this paper is to study the stability of FPUT phenomenon in the nonlinear fibers numerically. In this study, we take the sinusoidally perturbed continuous waves(CWs) with white noise imposed as the initial condition, which can be readily realized in real experiments. We find that both the perturbation amplitude and phase difference between the perturbation and pump can take drastic effects in affecting the stability of the resulting FPUT phenomenon. Firstly, as the perturbation amplitude increases, FPUT phenomenon becomes much more stable. When the perturbation amplitude reaches a critical value, the most stable FPUT phenomenon can be observed. As the further increase of the perturbation amplitude, the stability of the resulting FPUT phenomenon weakens. Secondly, the phase difference between the perturbation and pump take distinct effects on the stability of FPUT phenomenon for perturbation frequency locating in and outside of the conventional modulation instability (MI) band. For perturbation frequency locating inside the conventional MI band, as the phase difference between the perturbation and pump increases from zero, the corresponding FPUT phenomenon first becomes less stable; then most instable FPUT phenomenon can be observed at a critical phase difference; after then, the stability of the FPUT phenomenon enhances again. For perturbation frequency locating outside the conventional MI band, the stability of FPUT phenomenon enhances monotonously as the phase difference increases from 0 to π/2.To observe much more stable FPUT phenomenon, the above results show that the perturbation amplitude should be moderately large, and that the phase difference between the perturbation and the pump should be appropriate to avoid the most instable FPUT phenomenon that may arise.