Speaker recognition is a well known and studied task in the speech processing domain. It has many applications, either for security or speaker adaptation of personal devices. In this paper, we present a new paradigm for automatic speaker recognition that we call Interactive Speaker Recognition (ISR). In this paradigm, the recognition system aims to incrementally build a representation of the speakers by requesting personalized utterances to be spoken in contrast to the standard text-dependent or text-independent schemes. To do so, we cast the speaker recognition task into a sequential decision-making problem that we solve with Reinforcement Learning. Using a standard dataset, we show that our method achieves excellent performance while using little speech signal amounts. This method could also be applied as an utterance selection mechanism for building speech synthesis systems.