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The Ising model has been explored as a framework for modeling NP-hard problems, with several diverse systems proposed to solve it. The Magnetic Tunnel Junction– (MTJ) based Magnetic RAM is capable of replacing CMOS in memory chips. In this article, we propose the use of MTJs for representing the units of an Ising model and leveraging its intrinsic physics for finding the ground state of the system through annealing. We design the structure of a basic MTJ-based Ising cell capable of performing the functions essential to an Ising solver. The hardware overhead of the Ising model is analyzed, and a technique to use the basic Ising cell for scaling to large problems is described. We then go on to propose Ising-FPGA, a parallel and reconfigurable architecture that can be used to map a large class of NP-hard problems, and show how a standard Place and Route tool can be utilized to program the Ising-FPGA. The effects of this hardware platform on our proposed design are characterized and methods to overcome these effects are prescribed. We discuss how three representative NP-hard problems can be mapped to the Ising model. Further, we suggest ways to simplify these problems to reduce the use of hardware and analyze the impact of these simplifications on the quality of solutions. Simulation results show the effectiveness of MTJs as Ising units by producing solutions close/comparable to the optimum and demonstrate that our design methodology holds the capability to account for the effects of the hardware.