The ubiquitous presence of electronic devices demands robust hardware security mechanisms to safeguard sensitive information from threats. This paper presents a physical unclonable function (PUF) circuit based on magnetoresistive random access memory (MRAM). The circuit utilizes inherent characteristics arising from fabrication variations, specifically magnetic tunnel junction (MTJ) cell resistance, to produce corresponding outputs for applied challenges. In contrast to Arbiter PUF, the proposed effectively satisfies the strict avalanche criterion (SAC). Additionally, the grid-like structure of the proposed circuit preserves its resistance against machine learning-based modeling attacks. Various machine learning (ML) attacks employing multilayer perceptron (MLP), linear regression (LR), and support vector machine (SVM) networks are simulated for two-array and four-array architectures. The MLP-attack prediction accuracy was 53.61% for a two-array circuit and 49.87% for a four-array circuit, showcasing robust performance even under the worst-case process variations. In addition, deep learning-based modeling attacks in considerable high dimensions utilizing multiple networks such as convolutional neural network (CNN), recurrent neural network (RNN), MLP, and Larq are used with the accuracy of 50.31%, 50.25%, 50.31%, and 50.31%, respectively. The efficiency of the proposed circuit at the layout level is also investigated for simplified two-array architecture. The simulation results indicate that the proposed circuit offers intra and inter-hamming distance (HD) with a mean of 0.98% and 49.96%, respectively, and a mean diffuseness of 49.09%.