In the human brain, learning is continuous, while currently in AI, learning algorithms are pre-trained, making the model non-evolutive and predetermined. However, even in AI models, environment and input data change over time. Thus, there is a need to study continual learning algorithms. In particular, there is a need to investigate how to implement such continual learning algorithms on-chip. In this work, we focus on Oscillatory Neural Networks (ONNs), a neuromorphic computing paradigm performing auto-associative memory tasks, like Hopfield Neural Networks (HNNs). We study the adaptability of the HNN unsupervised learning rules to on-chip learning with ONN. In addition, we propose a first solution to implement unsupervised on-chip learning using a digital ONN design. We show that the architecture enables efficient ONN on-chip learning with Hebbian and Storkey learning rules in hundreds of microseconds for networks with up to 35 fully-connected digital oscillators.