The use of model order reduction techniques in combination with ensemble-based methods for estimating the state of systems described by nonlinear partial differential equations has been of great interest in recent years in the data assimilation community. Methods such as the multi-fidelity ensemble Kalman filter (MFEnKF) and the multi-level ensemble Kalman filter (MLEnKF) have been developed and implemented in several papers and are recognized as state-of-the-art techniques. However, the construction of low-fidelity models in the offline stage, prior to solving the data assimilation problem, leads these methods into a trade-off between the accuracy and computational cost of the approximate models. In this work, we investigate the use of adaptive reduced-basis techniques in which the approximation space is modified (but not retrained) online based on the information extracted from the full-order solutions. This has the potential to simultaneously ensure good accuracy and low cost for the employed models and thus improve the performance of the multi-fidelity/multi-level methods.