Computational power is a challenge when it comes to the high-fidelity modeling of nuclear reactors. Detailed simulations of reactor physics involve complex calculations that require significant computing resources, which can be time-consuming and expensive. Reduced Order Modeling (ROM) allows for an approximation of a complex model by only capturing the essential features, thereby reducing the computational load. A reduced order model provides computationally efficient approximations of a system, but it requires still many evaluations of a high-fidelity model to capture all the dynamics. Using the adaptive sparse grid can reduce the number of evaluations needed, though the construction of the reduced order model is still computationally intensive. The aim is to minimize the computational workload involved in constructing a reduced-order model during the offline phase. This is achieved by decreasing the number of high-fidelity model evaluations necessary for building the reduced order model while maintaining accurate results. To this end, the existing adaptive proper orthogonal decomposition algorithm is enhanced by employing multi-fidelity techniques. Multi-fidelity methods aim to combine large amount of low-fidelity data with a limited amount of high-fidelity data to compute accurate, yet computationally inexpensive approximations. Two novel multi-fidelity reduced order model methods based on proper orthogonal decomposition are proposed; Filtered Bi-Fidelity Adaptive Proper Orthogonal Decomposition (FB-POD) algorithm and Adapted Bi-Fidelity Proper Orthogonal Decomposition (AB-POD). These models are evaluated on two different test cases, and the balance between the accuracy of each multi-fidelity ROM and the computational cost, measured by the number of high-fidelity evaluations, is investigated. In specific cases, the proposed methods significantly reduce the number of high-fidelity evaluations compared to the single high-fidelity ROM, while yielding comparable accuracy.