During the LHC Run-I (2010–2013) and Run-II (2015–2018), the selection of interesting events for muon physics in ALICE was performed with a dedicated muon trigger system, composed of 72 single-gap phenolic Resistive Plate Chambers (RPCs) operated with current front-end electronics (ADULT FEE) that provides no amplification for the detector pulses. From Run-III starting in 2021 on, in order to fully profit from the increased luminosity of Pb-Pb collisions, the ALICE experiment will run in continuous readout (triggerless) mode and the Muon TRigger system (MTR), therefore, plays the role of a Muon IDentifier (MID). The read-out electronics is being upgraded in order to support continuous readout. Furthermore, in order to increase the RPC rate capability and to mitigate possible aging effects, it is advantageous to operate the detectors with a lower gain avalanche mode by using higher sensitive FEE. Therefore, we decided to replace the current FEE cards with new ones named Front-End Electronics Rapid Integrated Circuit (FEERIC) equipped with an amplification stage. Also, an upgrade of the threshold distribution system to the front-end will allow one to tune thresholds at the single front-end card level, while this was previously only possible at the single-RPC level. Finally, since some of the RPCs currently installed in ALICE have accumulated charges significantly comparable to the value that is expected in their lifespan, roughly 25% of the detectors are planned to be replaced by new ones constructed with new phenolic high-pressurized laminates (HPL). A detailed description of the MTR upgrade and of its current status will be presented in this contribution.