A two-step inexact Newton-Chebyshev-like method is proposed to solve inverse eigenvalue problems. Unlike general Newton type methods which need to invert the Jacobian matrix that may lead to an unexpected instability, the proposed method can guarantee the numerical stability by exploiting the Chebyshev method to approximate the inverse of the Jacobian matrix. Theoretically, we prove that the proposed method converges cubically. Numerical examples are given to demonstrate the effectiveness of our method.