Identifying dynamical system models from measurements is a central challenge in the structural dynamics community. Nonlinear system identification, in particular, is a big challenge as there are many possible combinations of model structures, which requires expert knowledge to construct an appropriate model. Furthermore, traditional nonlinear system identification methods require a steady excitation input that is not always available in many practical applications. Recently, a technique referred to as the sparse identification of nonlinear dynamics (SINDy) algorithm was developed to discover mathematical models of general nonlinear systems. The SINDy method finds a generalized linear state-space model for an autonomous nonlinear system by analyzing the collected response data. In this work, the SINDy method is adapted and combined with the shooting method and the numerical continuation technique to form a system identification framework that can predict the nonlinear modal properties of mechanical oscillators. The proposed framework predicts the nonlinear normal modes (NNMs) of these systems by processing the noised data of the systems' free vibration response. In addition, the NNMs and the internal resonance of nonlinear systems at a high energy level can be captured using the proposed framework by processing the response data at a lower energy level. The proposed method is numerically demonstrated on a 2-degree-of-freedom mechanical oscillator. Furthermore, the effects of the measurement error and the excitation condition on NNM prediction are investigated. The NNM prediction framework presented in this paper is fairly general and is applicable to a variety of nonlinear systems.