Assume that we have a priori information that a nonlinear system satisfies a physical property that makes it easy to design a controller and obtain a desired performance or stability guarantee on the closed loop system. Can we identify a system model that satisfies this property? In particular, we consider the property to be that of dissipativity. Dissipativity is an important input-output property of dynamical systems which encompasses many important special cases like L2 stability, passivity and conicity. Dissipativity, thus, finds application in various domains ranging from robotics, electromechanical systems, and aerospace systems to process control, networked control, cyber-physical systems, and energy. To address this problem, we will discuss a two-stage approach where we first learn an approximate linear models of the nonlinear system using classical or Koopman operator approaches, and then perturb the system matrices of the linear model to enforce dissipativity, while closely approximating the dynamical behavior of the nonlinear system. Further, we provide an analytical relationship between the size of the perturbation and the radius in which the dissipativity of the linear model guarantees local dissipativity of the unknown nonlinear system. We demonstrate the application of this identification technique to the problem of learning dissipative models of power systems.